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# Dynamical features of neurons and the brain: Chaos, synchronization, and propagation

### From DynamicBrain Platform

Hatsuo Hayashi
Kyushu Institute of Technology

## 1. Introduction

There are many ways of studying the nonlinear dynamics of neurons and the brain, and I myself have also took several approaches in studying neurons and the brain from a nonlinear dynamics point of view.

Since neurons can be represented by a conductance-based model such as the Hodgkin-Huxley model [Hodgkin and Huxley, 1952], they can be considered to be a deterministic dynamical system. However, as this model is extremely nonlinear, its behaviour was not sufficiently well understood until the 1970s. At the time it was known that neuronal activity was entrained by periodic stimuli as a typical nonlinear phenomenon [Holden, 1976; Guttman et al., 1980], but as for the other irregular neuronal activities, nobody had known even whether they were nonlinear phenomena or not.

If excitatory cells (whether plant or nerve cells) are stimulated electrically, they will soon produce various irregular responses that we are not sure of how to explain, besides regular responses such as phase-lockings. Although nowadays many of such irregular responses can be understood as chaos, it could not be denied previously that such irregular responses were no more than effects of noise. If this were the case, the problem would be to find the source of the noise, and, for example, the opening and closing of ion channels was thought to be a candidate. However, we could not accept such microscopic source of noise, because the irregular activity was produced as if neurons were affected by a large amount of external noise in spite of observing activity of isolated single neurons. If the Hodgkin-Huxley model were correct, the neuron would be a deterministic dynamical system. It was not easy to appreciate intuitively that random behaviour at the microscopic level has a tremendous effect on deterministic behaviour at the macroscopic level. It would be natural to consider that macroscopic, irregular responses are rather produced due to an inherent nonlinear nature of the neuron regardless of noise.

The discovery of chaos in molluscan neurons [Hayashi et al., 1982, 1983, 1985, 1986] gave an impetus to research on neurodynamics, and research on chaos using neuron models was also conducted actively in the 1980s. The reader should refer to the book [Hayashi (2001)] for the leading researches in this area. What is more, research on chaos in neurons led to the discovery of chaos in the brain [Hayashi and Ishizuka, 1995; Ishizuka and Hayashi, 1996], and provided a base to understand brain activity in the framework of nonlinear dynamics. Nowadays we are able to understand neuronal activity, including irregular activity, as deterministic dynamical phenomena, and are also able to control bifurcations between phase-lockings and chaos by altering suitable bifurcation parameters. Noise is not essential to causing chaos, though chaos is affected sensitively by noise.

The relationship between brain functions and nonlinear phenomena of neurons and neural networks has not been well understood, and still remains to be resolved. As described in the next section and thereafter, the membrane potential and the local field potential were recorded when chaos in neurons and neural networks were observed, respectively. In order to examine the relationships between chaos and brain functions it may be necessary to record more macroscopic variables. In this sense, it is desired that the resolution of a non-invasive apparatus like f-MRI, which is able to measure brain activity over a wide range of the brain, is improved more. Another candidate is the computer having a large data processing capacity. It may be an extreme way to create a realistic brain model using a lot of neurons represented by deterministic dynamical equations, but with rapid advances being made in computer processing capacity, this may not necessarily be so. Even using computers available in research laboratories today, it is possible to carry out a computer simulation using a neural network model consisting of 103-105 neurons. Even if it is difficult to execute a computer simulation using a model of the whole brain, it would be possible to develop a realistic model of a certain functional local area in the brain and carry out a computer simulation. If this were possible, it would not only lead to rapid advances in our understanding of information processing mechanisms in the brain but also greatly help in developing brain-like information processing systems.

Rreferences

1. Hodgkin AL and Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117: 500-544.
2. Holden AV (1976) The response of excitable membrane models to a cyclic input. Biol. Cybern. 21: 1-7.
3. Guttman R, Feldman L, and Jakobsson E (1980) Frequency entrainment of squid axon membrane. J. Memb. Biol. 56: 9-18.
4. Hayashi H, Ishizuka S, Ohta M, and Hirakawa K (1982) Chaotic behavior in the Onchidium giant neuron under sinusoidal stimulation. Phys. Lett. 88A: 435-438.
5. Hayashi H, Ishizuka S, and Hirakawa K (1983) Transition to chaos via intermittency in the Onchidium pacemaker neuron. Phys. Lett. 98A: 474-476.
6. Hayashi H, Ishizuka S, and Hirakawa K (1985) Chaotic response of the pacemaker neuron. J. Phys. Soc. Japan 54: 2337-2346.
7. Hayashi H, Ishizuka S, and Hirakawa K (1986) Instability of harmonic responses of Onchidium pacemaker neuron. J. Phys. Soc. Japan 55: 3272-3278.
8. Hayashi H (2001) "Chaos in the Brain" Shokabo [in Japanese].
9. Hayashi H and Ishizuka S (1995) Chaotic responses of the hippocampal CA3 region to a mossy fiber stimulation in vitro. Brain Res. 686: 194-206.
10. Ishizuka S and Hayashi H (1996) Chaotic and phase-locked responses of the somatosensory cortex to a periodic medial lemniscus stimulation in the anesthetized rat. Brain Res. 723: 46-60.
(Dec. 25, 2014)

## 2. Chaotic Response of the Onchidium Giant Silent Neuron

Since absolute and relative refractory periods follow the generation of each action potential in neurons, applying a high-frequency impulse train changes the form and the latency of the action potential frequently. Therefore, even in a single neuron, responses are often irregular, and it has been demonstrated that the irregular responses are nonlinear phenomena, which can be understood on the basis of deterministic nonlinear dynamics. Such irregular responses, which seem random at first glance, are nonperiodic oscillations referred to as chaos.

In excitable cells, chaos had been demonstrated in cardiac muscle cells [Guevara et al. 1981] and plant Nitella flexilis internodal cells [Hayashi et al. 1982a], and evidence for chaotic response of neurons was provided for the first time in 1982 [Hayashi et al. 1982b]. The giant silent neuron in the esophageal ganglion of the mollusk Onchidium verruculatum (Fig. 2.1) was used in the experiments. Chaotic membrane potential responses to sinusoidal current stimulation were recorded using a glass microelectrode.

Fig. 2.1  Mollusk Onchidium verruculatum caught on the seashore of the Sakura-jima island, Kagoshima prefecture, Japan.

We can reconstruct an attractor from the chaotic membrane potential response in a two-dimensional phase plane having the membrane potential V and its time derivative dV/dt as the axes (Fig. 2.2). Trajectories in Fig. 2.2 (a) and the phase portrait of the attractor in Fig. 2.2 (b) were obtained from the same response. In Fig. 2.2 (a), trajectories can be seen as the response was sampled at a high rate over a short period of time. In Fig. 2.2 (b), since the response was sampled at a low rate over a long period of time, the phase portrait of the attractor can be observed. This attractor is called the strange attractor, and is the first evidence for chaotic response of single neurons [Hayashi et al. 1982b].

Fig. 2.2  Trajectories and the strange attractor obtained from a chaotic response of the giant silent neuron in the mollusk Onchidium verruculatum. Those are obtained from the same data, and projected onto a two-dimensional phase plane (V – dV/dt). (a) Trajectories. As the response was sampled at a high rate over a short period of time, trajectories can be seen [Hayashi, 2001]. (b) Phase portrait of the strange attractor. As the response was sampled at a low rate over a long period of time, the phase portrait can be seen [Hayashi et al., 1982b].

It is the most certain way to investigate the geometric structure of the attractor in order to show that the attractor in Fig. 2.2 (b) is a strange attractor. In these experiments, since a sinusoidal current stimulation was applied to the neuron, it is a good way to use the three-dimensional phase space (ΦV – dV/dt) having the phase Φ of the sinusoidal current as an additional axis (Fig. 2.3). Schematic trajectories are superimposed in the range of the phase 0 < Φ < 2π (Fig. 2.3 (a)), and the attractor is represented schematically in Fig. 2.3 (b). A torus is obtained by bending the Φ axis into a curve and attaching two vertical planes (Φ = 0 and Φ = 2π) to each other.

Fig. 2.3  Trajectories and the strange attractor reconstructed from the chaotic response of the giant neuron in the Onchidium verruculatum. Those are represented schematically in a three-dimensional phase space (ΦV – dV/dt). (a) Trajectories. Intersections {P1, P2, ---} of the plane S and the trajectories are also shown. (b) Strange attractor. Cross-sections, a-i, of the attractor are also shown. [adapted from Hayashi, 2001]

The set of intersections {P1, P2, P3, ---} of the vertical plane S and trajectories shown in Fig. 2.3 (a) provides a cross-section of the attractor. Cross-sections obtained at 9 different vertical planes are shown schematically in Fig. 2.3 (b) (a-i). It is easy to obtain such a cross-section from the chaotic response recorded experimentally. It can be obtained by sampling the response at a phase Φ every period of the sinusoidal current. Figure 2.4 shows cross-sections obtained from the experimental data and projected onto the two-dimensional plane (V – dV/dt).

Fig. 2.4   Cross-sections of the strange attractor obtained from the chaotic response of the Onchidium giant silent neuron. The stretching (b-d) and folding (e-h) of the attractor can be seen. Broken lines indicate the outline of the strange attractor. Phases Φ of the sinusoidal current in (a)-(i), at which the response was sampled, are indicated in (j). [Hayashi et al., 1982b]

Since such cross-sections are almost one-dimensional, the strange attractor is flat and almost two-dimensional (Fig. 2.3 (b)). This is because the attractor is stable in the normal direction of the attractor (negative eigenvalue), so that nearby trajectories are attracted towards the attractor in the normal direction, and unstable in the tangential direction of the attractor (positive eigenvalue), so that the distance between trajectories increases exponentially in the tangential direction. This is referred to as the hyperbolicity of the strange attractor (Refer to Section 2.2 in Hayashi, 2001, for more details). The exponent averaged over the whole attractor is called the Lyapunov exponent, and indicates the sensitivity to initial conditions.

In Fig. 2.4, the cross-section is progressively stretched ((b) – (d)) and folded ((e) – (h)). This is referred to as the stretching and folding of the attractor. As the stretching and folding are repeated every cycle of the sinusoidal current, trajectories are mixed with time within the attractor. Consequently, the trajectories are getting complex and unpredictable. This is the mechanism for chaos generation, and an attractor like that is called the strange attractor. Such irregular responses of neurons due to the stretching and folding of the attractor are not random but deterministic.

Fig. 2.5  One-dimensional map F: VnVn+1 obtained from the chaotic response of the Onchidium giant silent neuron. Φ = 150°. Panels, a and b, show the same data. (a) V0 is the fixed point because it satisfies V0 = F(V0). (b) Blue lines indicate iteration of the mapping. [adapted from Hayashi et al., 1982b]

A one-dimensional map F: VnVn+1 (Fig. 2.5) can be obtained from V-coordinates of the intersections {P1, P2, P3, ---} of the plane S and the trajectories in Fig. 2.3 (a). Since the intersection V0, which is also the intersection of the map F and the diagonal line, satisfies V0 = F(V0), the intersection V0 is a fixed point (Fig. 2.5 (a)). However, it is unstable, because the slope of the map F at V0 is more negative than -1. Therefore, if mapping is iterated from the initial intersection Vint in the vicinity of the unstable fixed point V0 as shown in Fig. 2.5 (b), the intersection obtained every mapping goes away from the unstable fixed point. As the map is a convex function, the intersection is pushed back frequently to the vicinity of the unstable fixed point, and goes away from the unstable fixed point again. Therefore, the intersection wanders around on the two-dimensional plane (Vn - Vn+1). In other words, the trajectory passing through the intersection Pint = (Vint, dVint/dt) goes away rapidly from the unstable periodic trajectory that passes through the unstable fixed point P0 = (V0, dV0/dt) in the three-dimensional phase space (ΦV – dV/dt). Then, the trajectory is pushed back frequently to the vicinity of the unstable periodic trajectory due to the folding of the attractor, and goes away again. Going away from the fixed point due to instability and being pushed back to the vicinity of the unstable fixed point due to convexity of the one-dimensional map correspond to the stretching and the folding of the strange attractor, respectively.

As mentioned above, chaotic response of a single neuron is generated by a very simple dynamical rule, namely the one-dimensional map, and bifurcation phenomena have also been elucidated as mentioned in the following chapters. Such neuro-chaos discovered in the experiments stimulated people to study it using neuron models, and features of the neuro-chaos and bifurcation phenomena have been investigated in detail. The readers should refer to the book [Hayashi, 2001] for leading works on chaos and bifurcation phenomena using neuron models. The results obtained in experiments and computer simulations gave impetus to study of neurodynamics.

References

1. Guevara MR, Glass L, and Shrier A (1981) Phase locking, period-doubling bifurcation, and irregular dynamics in periodically stimulated cardiac cells. Science 214: 1350-1353.
2. Hayashi H, Nakao M, and Hirakawa K (1982a) Chaos in the self-sustained oscillation of an excitable biological membrane under sinusoidal stimulation. Phys. Lett. 88A: 265-266.
3. Hayashi H, Ishizuka S, Ohta M, and Hirakawa K (1982b) Chaotic behavior in the Onchidium giant neuron under sinusoidal stimulation. Phys. Lett. 88A: 435-438.
4. Hayashi H (2001) "Chaos in the Brain" Shokabo [in Japanese].
(Mar. 17, 2015)

## 3. Chaotic Responses of the Onchidium Pacemaker Neuron

Chaotic response of the giant silent neuron in the oesophagel ganglion of Onchidium verruculatum was shown in the previous chapter. A pacemaker neuron, which fires repeatedly without stimulation in the artificial sea water, also exists in the same ganglion. In this chapter, I will show phase-locked and chaotic responses of the pacemaker neuron to periodic current stimulation, and besides, bifurcation phenomena.

The abscissa and the ordinate of the phase diagram in Fig. 3.1 are the frequency and the intensity of sinusoidal current stimulation, respectively. Regions of phase-locked responses (1:1, 1:2, and 1:3) and chaotic responses (★, ▲, ●, ■, and ☆) exist in the phase diagram, and we can see that bifurcations occur depending on the frequency and the intensity of the periodic stimulation. More complex phase-locked responses, such as 3:4, 2:3, and 3:5 phase-locked responses, occur between the regions of the 1:1 and 1:2 phase-locked responses, as shown in Fig. 3.2, and similar complex phase-locked responses are also observed between the regions of the 1:2 and 1:3 phase-locked responses.

Fig. 3.1   Phase diagram of membrane potential response of the Onchidium pacemaker neuron to sinusoidal current stimulation. fi: Frequency of sinusoidal current stimuation. f0: Frequency of spontaneous firing of the pacemaker neuron. SR: Amplitude of subthreshold response to sinusoidal current stimulation. AP: Amplitude of the action potential. ★, ▲, ●, ■, and ☆: Chaotic responses. ○(1:1), △(1:2), and □(1:3): Phase-locked responses, in which one action potential is elicited every one, two, and three cycles of the sinusoidal current stimulation, respectively. (Although the chaos ☆ is referred to as the random alternation in Hayashi et al., 1985, it has been demonstrated that the random alternation is not random but chaotic in Hayashi et al., 1986.) [adapted from Hayashi et al., 1985]

Fig. 3.2   Various phase-locked responses of the Onchidium pacemaker neuron caused by sinusoidal current stimulation between the regions of the 1:1 and 1:2 phase-locked responses in Fig. 3.1. For example, 4/5 represents the 4:5 phase-locked response. [Hayashi and Ishizuka, 1993]

The pacemaker neuron causes several kinds of chaotic responses depending on the frequency and the intensity of the sinusoidal current stimulation. Three kinds of chaotic membrane potential responses (★, ▲, and ☆) and strange attractors reconstructed from those responses are shown in Fig. 3.3 and Fig. 3.4, respectively. It should be noted that the three-dimensional phase space for reconstructing the attractors does not have the phase of the sinusoidal current as an axis, but the current itself I. I here focus on the chaotic response marked by ★ on account of limited space. Readers should refer to the articles [Hayashi et al., 1985, 1986] for the other chaotic responses.

Fig. 3.3   Three kinds of chaotic membrane potential responses of the Onchidium pacemaker neuron. The upper and lower traces in each panel are the membrane potential response and the sinusoidal stimulus current, respectively. [adapted from Hayashi et al., 1985]

Fig. 3.4   Strange attractors reconstructed from three kinds of chaotic responses of the Onchidium pacemaker neuron shown in Fig. 3.3. P1 and P2: Trajectories corresponding to action potentials. P3: Trajectories corresponding to subthreshold responses. [adapted from Hayashi et al., 1986]

In the same manner mentioned in Chapter 2, a cross-section of the strange attractor can be obtained by sampling the chaotic response (★) every period of the sinusoidal stimulus current at a phase of the current, and plotting the sampled membrane potentials on the two-dimensional space (V - dV/dt) (Fig. 3.5). It can be seen that the strange attractor is stretched (Fig. 3.5 (a)-(c)) and then folded (Fig. 3.5 (d)-(i)) with increase in the phase of the sinusoidal stimulus current. We can also obtain a one-dimensional map from the time series of sampled membrane potentials, as shown in Fig. 3.6. The fixed point of the one-dimensional map, which is the intersection of the map and the diagonal line, is unstable. Therefore, a sampled membrane potential in the vicinity of the unstable fixed point goes away from the fixed point with iteration of mapping. However, as the one-dimensional map is a convex function, the sampled membrane potential is pushed back to the vicinity of the unstable fixed point, and then goes away again. These results demonstrate that the chaotic response (★) follows a deterministic dynamical rule.

Fig. 3.5   Cross-sections of the strange attractor obtained from the chaotic response (★) of the Onchidium pacemaker neuron. Cross-sections are projected onto the two-dimensional plane (V - dV/dt). Dotted lines indicate the outline of the strange attractor projected onto the plane. The cross-section is stretched (a - c) and folded (d - i) with increase in the phase of the sinusoidal stimulus current. Phases of the sinusoidal current in (a) - (i), at which the response was sampled, are indicated in (j). [Hayashi and Ishizuka, 1987]

Fig. 3.6   One-dimensional map obtained from the chaotic response (★) of the Onchidium pacemaker neuron. The phase of the sinusoidal stimulus current, at which the response was sampled, was 150°. [Hayashi et al., 1983]

As shown in Fig. 3.1, the Onchidium pacemaker neuron does not bifurcate from the 1:2 phase-locking (△) to the chaos (★) through a sequence of period-doubling bifurcations, but through the intermittency (▲). This is because the one-dimensional map has a sharp peak, as shown in Fig. 3.6 (Refer to Hayashi et al., 1985 for further details).

References

1. Hayashi H, Ishizuka S, and Hirakawa K (1983) Transition to chaos via intermittency in the Onchidium pacemaker neuron. Phys. Lett. 98A: 474-476.
2. Hayashi H, Ishizuka S, and Hirakawa K (1985) Chaotic response of the pacemaker neuron. J. Phys. Soc. Japan 54: 2337-2346.
3. Hayashi H, Ishizuka S, and Hirakawa K (1986) Instability of Harmonic responses of Onchidium pacemaker neuron. J. Phys. Soc. Japan 55: 3272-3278.
4. Hayashi H and Ishizuka S (1987) Chaos in molluscan neuron. in “Chaos in Biological Systems” Eds. Degn H, Holden AV, and Olsen LF, NATO ASI Series, A 138: 157-166, Plenum Press.
5. Hayashi H and Ishizuka S (1993) Complex activities of neurons and neural networks - chaos-. in "Chaos in Neural Systems" Ed. Aihara K, Chapter 1 (pp.1-48), Tokyo Denki University Press [in Japanese].
(April 14, 2015)

## 4. Spontaneous Chaotic Firing of the Onchidium Pacemaker Neuron

Chaotic responses of Onchidium silent and pacemaker neurons to periodic stimulation and bifurcation phenomena were shown in Chapters 2 and 3. I here show that the Onchidium pacemaker neuron causes chaotic firing spontaneously even in the absence of periodic stimulation.

Many of the pacemaker neurons cause burst firing, in which groups of action potentials are caused at relatively long intervals. This is ascribed to ion currents that produce a slow membrane potential oscillation. Repetitive firing occurs in depolarizing phases of the slow membrane potential oscillation, and stops in hyperpolarizing phases (Fig. 4.1). In other words, considering the pacemaker neuron as a dynamical system consisting of a fast subsystem, which causes action potentials, and a slow subsystem, which causes a slow membrane potential oscillation, bursts of action potentials occur due to interaction between the two subsystems (Refer to the article [Rinzel, 1987] and/or the book [Hayashi, 2001] for further details).

Fig. 4.1   Spontaneous burst firing of the Onchidium pacemaker neuron. Action potentials occur in depolarizing phases of the slow membrane potential oscillation, and stops in hyperpolarizing phases. Dashed line is a schema of the slow membrane potential oscillation. [adapted from Hayashi and Ishizuka, 1992]

Fig. 4.2   Spontaneous firing of the Onchidium pacemaker neuron. (a) Period 1. Idc = 0.96 nA. (b) Period 2. Idc = 0.41 nA. (c) Period 4. Idc = 0.26 nA. (d) Chaos. Idc = 0.185 nA. (e) Period 3. Idc = -0.11 nA. Idc is a dc current applied to the pacemaker neuron. Outward dc current (positive current) depolarizes the neuron. [adapted from Hayashi and Ishizuka, 1992]

As the slow subsystem, which causes a slow membrane potential oscillation, interplays with the fast subsystem, which causes action potentials, the pacemaker neuron produces various firing patterns depending on depolarization and hyperpolarization of the neuron. When the membrane potential is changed from a depolarized state to a hyperpolarized state, the firing pattern bifurcates from period 1 to period 3 through period 2, period 4, ---, and chaos, as shown in Fig. 4.2. On the route to the chaos from the period 1, the period of the firing pattern doubles successively when bifurcations take place. These bifurcations are called the period-doubling bifurcation.

Attractors reconstructed from spontaneous firing patterns in Fig. 4.2 (a), (b), (d), and (e) are shown in Fig. 4.3 (a)-(d), respectively. Attractors of the period 1, 2, and 3 show stable periodic trajectories (one, two, and three loops), respectively (Fig. 4.3 (a), (b), and (d)). However, chaotic trajectories do not converge in a periodic trajectory because the chaotic trajectories are unstable (Fig. 4.3 (c)). It can be demonstrated that this attractor is a strange attractor by the geometrical structure of the attracter showing stretching and folding characteristics and a one-dimensional map showing a convex function with an unstable fixed point.

Fig. 4.3   Attractors obtained from spontaneous firing of the Onchidium pacemaker neuron shown in Fig. 4.2 (a), (b), (d), and (e). Attractors were reconstructed in a three-dimensional space (V(t), V(t +τ), V(t + 2τ). τ = 4 ms. (a) Period 1. (b) Period 2. (c) Chaos. (d) Period 3. The inside loop of the period 3 consists of two trajectories. The plane S shown in (c) was used to obtain a cross-section of the attractor. [adapted from Hayashi and Ishizuka, 1992]

A cross-section of the strange attractor can be obtained as a set of intersections of the plane S and trajectories. The plane S was rotated around the z axis, which was parallel to the V(t + 2τ) axis, and the location of the plane S was defined by the angle between the r and V(t) axes (Fig. 4.4 (a)). Cross-sections obtained at nine different vertical planes are shown in Fig. 4.4 (b). These cross-sections indicate that the attractor is flat like a ribbon, and has hyperbolicity. Moreover, the attractor is stretched and folded (210-330°). Therefore, unstable trajectories are mixed, as shown in Fig. 4.5. Consequently, firing of the pacemaker neuron becomes irregular and unpredictable.

Fig. 4.4   Cross-sections of the strange attractor obtained from spontaneous chaotic firing of the Onchidium pacemaker neuron shown in Fig. 4.3 (c). (a) Attractor projected onto the plane (V(t), V(t +τ)). Each line o-r indicates the location of the plane S, which was used to obtain a cross-section. The location of the plane S is defined by the angle between the line o-r and the V(t) axis. (b) Cross-sections of the strange attractor at nine different planes S. The attractor is stretched and folded (210-330°). [adapted from Hayashi and Ishizuka, 1992]

Fig. 4.5   Schematic diagram of the stretching and folding of the strange attractor. Trajectories are going away from each other due to instability, and then, some of the trajectories approach each other due to stretching and folding of the attractor. Consequently, trajectories are mixed in a part of the attractor. [adapted from Hayashi, 1998]

One dimensional map can be obtained from time series of intersections of the plane S and trajectories. One dimensional maps of the period 1, the period 2, and the period 3 have one, two, and three clusters of intersections, respectively (Fig. 4.6 (a), (b), and (d)). However, one dimensional map of the chaotic firing is a convex function, as shown in Fig. 4.6 (c). As the fixed point, which is the intersection of the map and the diagonal line, is unstable, intersections in the vicinity of the unstable fixed point go away from the fixed point with iteration of mapping. Those intersections are often pushed back to the vicinity of the unstable fixed point due to convexity of the map, and then, the intersections go away again. In other words, chaotic trajectories do not converge in a periodic trajectory, but wander around forever.

Geometrical structure of the attractor shown in Fig. 4.4 and the one-dimensional map shown in Fig. 4.6 (c) demonstrate clearly that the irregular firing in Fig. 4.2 (d) is chaotic and the attractor in Fig. 4.3 (c) is a strange attractor.

Fig. 4.6   One-dimensional maps obtained from time series of intersections of the plane S (240°) and trajectories shown in Fig. 4.3. Spontaneous firing of the Onchidium pacemaker neuron. [adapted from Hayashi and Ishizuka, 1992]

Response of the pacemaker neuron to periodic stimulation bifurcates to chaos through intermittency, as shown in the previous chapter (Fig. 3.1). Spontaneous firing of the Onchidium pacemaker neuron, however, bifurcates from period 1 to chaos through a sequence of period-doubling bifurcations (Fig. 4.2). This is because the one-dimensional map shown in Fig. 4.6 (c) does not have a sharp peak, but is rather parabolic. Readers should refer to the article [May, 1976] for period-doubling bifurcations caused by a parabolic one-dimensional map.

By the way, a conductance-based model of the Onchidium pacemaker neuron based on physiological properties can be developed as follows.

\begin{align} \scriptstyle{C_\mathrm{m}\, \dot{V}}\, &\scriptstyle{=\, g_\mathrm{Na}\, m^3\, h\, (V_\mathrm{Na}\, -\, V)\, +\, g_\mathrm{K}\, n^4\, (V_\mathrm{K}\, -\, V)\, +\, g_\mathrm{Nas}\, m_\mathrm{s}\, h_\mathrm{s}\, (V_\mathrm{Na}\, -\, V)\, +\, g_\mathrm{Ks}\, n_\mathrm{s}\, (V_\mathrm{K}\, -\, V)} \\ &\scriptstyle{+\, g_\mathrm{Ki}\, n_\mathrm{i}\, (V_\mathrm{K}\, -\, V)\, +\, g_\mathrm{L}\, (V_\mathrm{L}\, -\, V)\, -\, I_\mathrm{p}\, +\, I_\mathrm{dc}} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad (4.1) \end{align}
\begin{align} \scriptstyle{\dot{u}\,=\,}{{\scriptstyle{\alpha _u\,-\,(\alpha _u\,+\,\beta _u)\,u}} \over {\scriptstyle{\lambda _u}}},\quad \quad u\, \in\, \left\{ {m,\, h,\, n,\, m_\mathrm{s},\, h_\mathrm{s},\, n_\mathrm{s},\, n_\mathrm{i}} \right\} \qquad \qquad \qquad \qquad \qquad \qquad \quad (4.2) \end{align}

The first and second terms on the right-hand side of Eq. 4.1 represent the fast Na+ and K+ currents, which generate action potentials, respectively. The third and fourth terms are the slow Na+ and K+ currents with large time constants respectively, and these currents produce a slow membrane potential oscillation as shown in Fig. 4.1. The 5th-8th terms of Eq. 4.1 are the anomalous inward rectifier K+, leak, pump, and dc currents, respectively. The dc current Idc is a bifurcation parameter here. Readers should refer to the article [Hayashi and Ishizuka, 1992] and the book [Hayashi, 2001] for more details about the pacemaker neuron model.

Fig. 4.7   Bifurcation diagram of the Onchidium pacemaker neuron model. The ordinate is the peak potential of the action potential. The abscissa is the dc current applied to the neuron model. The period 1 bifurcates to the chaos through a sequence of period-doubling bifurcations with decrease in Idc. [adapted from Hayashi and Ishizuka, 1992]

The bifurcation diagram shows that spontaneous firing of the Onchidium pacemaker neuron bifurcates from period 1 to chaos through a sequence of period-doubling bifurcations with decrease in the dc current Idc (Fig. 4.7). Firing patterns reproduced by this neuron model are shown in Fig. 4.8. In the same way mentioned above, a one-dimensional map can be obtained from the spontaneous chaotic firing of the pacemaker neuron model. The one-dimensional map obtained numerically (Fig. 4.9) well reproduces the one-dimensional map obtained from experimental data (Fig. 4.6 (c)).

Fig. 4.8   Spontaneous firing patterns of the Onchidium pacemaker neuron model. (a) Period 1. Idc = -1.2 nA. (b) Period 2. Idc = -1.8 nA. (c) Chaos. Idc = -2.34 nA. [adapted from Hayashi and Ishizuka, 1992]

Fig. 4.9   One-dimensional map obtained from spontaneous chaotic firing of the Onchidium pacemaker neuron model. [adapted from Hayashi and Ishizuka, 1992]

Although silent neurons determine whether they respond or not to an input signal, the spontaneous firing pattern of the pacemaker neuron changes depending on depolarization and hyperpolarization, as mentioned above. It is, therefore, suggested that pacemaker neurons change the firing pattern sensitively depending on the integrated synaptic potential produced by input signals, neuromodulators, and so on, and may generate new information being sent to other neurons.

References

1. Hayashi H and Ishizuka S (1992) Chaotic nature of bursting discharges in the Onchidium pacemaker neuron. J. Theor. Biol. 156: 269-291.
2. Hayashi H (1998) "Nonlinear Phenomena in Nervous Systems" Corona Publishing Co. Ltd. Tokyo [in Japanese].
3. Hayashi H (2001) "Chaos in the Brain" Shokabo [in Japanese].
4. May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261: 459-467.
5. Rinzel J (1987) A formal classification of bursting mechanisms in excitable systems. Lecture Notes in Biomathematics 71: 267-281, Springer-Verlag.
(May 15, 2015)

## 5. Chaotic Response of the Rat Hippocampal Slice

Chaotic response and spontaneous chaotic firing of single neurons were mentioned in the previous chapters. It was natural that the discovery of chaos in single neurons stimulated researcher's interest in the issue whether irregular brain activities are chaotic. If irregular activities of the brain were chaotic, most brain activities could be understood in the framework of the deterministic nonlinear dynamics, and such understanding of the brain activities would also be important for investigating brain functions. In the present chapter, I will show how we can observe chaotic activities in the brain and what features the chaotic brain activities have.

Although research on chaotic activity of single neurons made a remarkable progress in 1980s, evidence for chaotic activity of the brain was not readily provided. I think the cause of this difficulty is that the brain activities are produced by neuronal assemblies, which consist of a huge number of neurons interacting with each other through excitatory and inhibitory connections. The brain activity is not a simple superposition of periodic and/or chaotic firings of individual neurons. Even if firings of individual neurons could be observed in detail, it would not be easy to understand dynamical features of brain activity, just as it is difficult to understand the dynamics of human mass activity even if characteristics of isolated individual persons are defined.

Researchers originally focused on the "low-dimensional" chaos. It was a great surprise and interest for them that deterministic nonlinear dynamical systems cause complex and unpredictable behaviors even if those dynamical dimensions are very low (∼ a few dimensions). However, neuronal assemblies consisting of many neurons are, in general, quite high dimensional systems.

Almost all the researchers used human electroencephalogram (EEG) data recorded from the scalp in order to provide evidence for chaotic brain activity. Although EEG indeed reflects a brain activity, EEG reflects activities of neuronal assemblies spreading in a wide range of the brain. In other words, activities of a huge number of neurons are observed simultaneously. It seems to be contradictory that they tried to provide evidence for "low-dimensional" chaotic activity in a very high-dimensional dynamical system. Moreover, EEGs are unsteady activities, which reflect brain states varying from moment to moment. These facts made EEG analyses difficult in terms of low-dimensional chaos. Researches are strange people. They carried out studies believing that the brain activity was chaotic. However, it would be important for researchers to have seemingly ridiculous belief even in the field of science regarding logic as important.

Babloyantz and her colleagues were the first to suggest that chaotic activity may exist in the brain [Babloyantz et al., 1985]. They showed that the correlation dimension (a kind of dynamical dimension) of human sleep EEG was very low (a few dimensions). It was a great surprise, and I think their results aroused many researchers to study of chaos in the brain. However, it was unfortunately demonstrated around 1990 that there is a risk of obtaining wrong dimensions when the algorithm for obtaining the correlation dimension, namely Grassberger-Procaccia algorithm (1983), is applied to EEGs [Osborne and Provenzale, 1989; Rapp et al., 1993]. Correlation dimensions of band-pass filtered noises are actually low non-integers; consequently, the noises seem to be chaotic. It should be note that EEGs are usually observed through a band-pass filter. So, reliability of many papers arguing chaotic brain activities based on the low correlation dimensions was questioned.

After that, the surrogate algorithm was developed to use the Grassberger-Procaccia algorithm properly for EEG analyses [Pijn et al., 1991; Theiler et al., 1992]. This algorithm arrows us to argue that EEGs are not necessarily random. However, the results obtained by the surrogate algorithm are very weak evidence for chaotic brain activity; we could rather say that the results do not provide any evidence for chaos in the brain.

Neural networks cause complex spatio-temporal activities, because neurons interact with each other through excitatory and inhibitory connections [Tateno et al., 1998]. However, these neurons do not fire inconsistently, but tend to fire in synchrony [Boddeke et al., 1997; Laurent, 1996; Whittington et al., 1995]. The fact that brain activities can be observed from the scalp as EEGs tells us so. As synchronization of neuronal activities increases the spatial coherence of the neural network, it might be possible that mass activity of the neuronal assembly is represented using a relatively small number of dynamical variables.

The amplitude of healthy EEGs is not so large. This implies that neuronal activities in the whole or partial brain are not completely synchronized. Probably, neuronal activities are synchronized here and there in local areas of the brain in a short period of time. This suggests that dynamical dimensions of EEGs are not sufficiently low to show features of low-dimensional chaos.

There would be several situations, in which synchronization of neuronal activities is enhanced and sustained. One is the case, in which rhythmic activities such as theta and delta rhythms occur in some region of the brain. Another one is the case, in which synchronization of neuronal activities is enhanced by periodic input. These situations allow us to expect that the dynamical dimension of local activity of the brain is much lower than that of EEGs.

We were trying to obtain indisputable evidence for chaos in the brain around the end of 1980s. After plenty of discussion, we decided to try it using the hippocampal CA3 and the somatosensory cortex of the rat. As the hippocampus is located under the neocortex, we used sagittal slices of the hippocampus. On the other hand, as activities of the somatosensory cortex can be recorded by electrode placed on the surface of the cortex, we used anesthetized rats (Chaotic responses of the somatosensory cortex will be shown in Chapter 8).

Neurons, which are connected to each other through rich recurrent connections, cause active firings spontaneously in the CA3 region of the hippocampus, and the CA3 region also causes theta rhythm, which is related to functions of memory and recognition. These suggest that neuronal activities are well synchronized in CA3. Moreover, since pyramidal cells, which are principal cells in the hippocampus, are parallel to each other in the pyramidal cell layer, synchronized activity of the pyramidal cells can be recorded as a field potential by a glass microelectrode placed on the layer (Fig. 5.1).

However, spontaneous rhythmic activity hardly occurs in hippocampal CA3 slices perfused by an artificial cerebrospinal fluid (ACSF). In order to cause synchronized spontaneous burst firings in hippocampal CA3 slices, GABAA inhibition was reduced by penicillin, and pyramidal cells were depolarized a little bit by increasing the K+ concentration of the medium.

Fig. 5.1   Sagittal plane of the rat hippocampus and the arrangement of electrodes (schematic diagram). Mossy fibers were stimulated by a bipolar electrode, and field potentials were recorded from the CA3 pyramidal cell layer using four extracellular glass microelectrodes (A-D). Intervals between electrodes are about 250 μm. [adapted from Hayashi and Ishizuka, 1995]

Fig. 5.2   Spontaneous epileptiform bursts caused in the CA3 region of the rat hippocampus. Penicillin (2 mM) was included in the artificial cerebrospinal fluid (ACSF). The K+ concentration in ACSF was 8 mM. (a) Field potential recorded by the electrode C in Fig. 5.1. Interburst intervals are about 3 sec (0.33 Hz). (b) Magnification of the trace in (a). The duration of the epileptiform burst is about 40 ms. Fast field potential variations reflect spikes of pyramidal cells. (c) Epileptiform burst smoothed by a low-pass filter (25 Hz cutoff). [adapted from Hayashi and Ishizuka, 1995]

Although spontaneous epileptiform bursts recorded by a glass microelectrode look like spikes (Fig. 5.2a), those are slow changes in the field potential (about 40 ms duration), on which fast field potential variations caused by spikes of pyramidal cells are superposed (Fig. 5.2b). Epileptiform bursts reflect the mass activities of a pyramidal cell assembly, namely partial synchrony of neuronal activities in the hippocampal network. Larger amplitude of the epileptiform burst indicates that more pyramidal cells fire in synchrony. Therefore, epileptiform bursts were smoothed by eliminating fast field potential variations using a low-pass filter in order to investigate dynamical features of mass activity of the pyramidal cell assembly (Fig. 5.2c).

Responses of a CA3 pyramidal cell assembly to synaptic inputs can be observed by stimulating mossy fibers. A bipolar electrode was used for mossy fiber stimulation, and field potential responses were recorded from the CA3 pyramidal cell layer using four extracellular glass microelectrodes (Fig. 5.1). Since a bundle of mossy fibers is stimulated, a stronger stimulation excites more fibers, and feeds a larger number of synaptic inputs into CA3.

Fig. 5.3   Field potential response of the hippocampal CA3 to periodic mossy fiber stimulation. The ordinate and the abscissa are the intensity and the frequency of stimulus current pulses, respectively. T is the interpulse interval. 1:n (n = 1, 2, 3, and 4): Phase-locked responses. One epileptiform burst occurs every n stimulus current pulses. Chaos: Chaotic responses. ☆: Random responses. In regions marked with ▲, ■, and ◆, complex phase-locked responses and irregular responses occur. [Hayashi and Ishizuka, 1995]

The phase diagram of field potential response of the hippocampal CA3 pyramidal cell layer is shown in Fig. 5.3. The ordinate and the abscissa of the phase diagram are the intensity and the frequency of stimulus current pulses, respectively. The phase diagram shows that phase-lockings and chaotic responses are caused by the mossy fiber stimulation depending on the stimulus intensity and frequency. For example, 1:1 phase locking bifurcates to 1:3 phase-locking through 1:2 phase-locking and chaos with increase in the stimulus frequency along the arrow. Although a sequence of period-doubling bifurcations is expected to occur between the 1:2 phase-locking and the chaos, it has not been observed. Since the one-dimensional map of the chaotic response has a sharp peak as mentioned below, it is supposed that period-doubling bifurcations do not occur, or they occur in a very narrow region of the stimulus frequency.

1:1 phase-locked and chaotic responses of the hippocampal CA3 are shown in Fig. 5.4a and 5.4b, respectively. The lower and upper traces in Fig. 5.4a(i) and 5.4b(i) are periodic current pulses to stimulate mossy fibers and field potential responses (epileptiform bursts) recorded from the hippocampal CA3 pyramidal cell layer, respectively. These field potential responses are smoothed by a low-pass filter. Identical field potential responses are caused by stimulus current pulses in the 1:1 phase-locking. On the other hand, the amplitude of the field potential response caused by periodic stimulation varies irregularly in the chaotic response.

Superposition of ten successive field potential responses, which are magnified, shows that the shapes of the field potential responses are the same in the 1:1 phase-locking (Fig. 5.4a(ii)). In the case of the chaotic response, the amplitude of field potential response varies violently; this indicates that the number of fired pyramidal cells and the degree of synchronization vary every stimulus (Fig. 5.4b(ii)).

Fig. 5.4   Field potential responses of the hippocampal CA3 to periodic mossy fiber stimulation. (a) 1:1 phase-locking. (b) Chaotic response. (i) Lower and upper trances are current pulses for mossy fiber stimulation and field potential responses, respectively. The field potential responses are smoothed by a low-pass filter. (ii) Magnification of the traces in (i). Ten successive epileptiform bursts are superposed in the upper traces. The lower traces are superposition of the current pulses. (iii) Attractors reconstructed in the three-dimensional space (V(t), V(t + τ), V(t + 2τ)). τ = 10 ms. (iv) One-dimensional maps obtained from a sequence of field potentials sampled every period of the mossy fiber stimulation. [(i), (iii), and (iv): adapted from Hayashi and Ishizuka, 1995; (ii): adapted from Hayashi and Ishizuka, 1997]

Attractors reconstructed in the three-dimensional space (V(t), V(t + τ), V(t + 2τ)) are shown in Fig. 5.4(iii). V is the field potential, and τ is 10 ms. In the 1:1 phase-locking, as the amplitude and the shape of the field potential response are always identical, the trajectory forms one stable loop in the three-dimensional space (Fig. 5.4a(iii)). In the chaos, as the amplitude of the field potential response varies irregularly, the trajectory does not converge into a loop. Consequently a strange attractor is reconstructed (Fig. 5.4b(iii)).

One-dimensional map can be obtained from a sequence of field potentials sampled every period of mossy fiber stimulation. In the case of 1:1 phase-locking, sampled field potentials cluster around a stable fixed point on the diagonal line of the map (Fig. 5.4a(iv)). On the other hand, one-dimensional map obtained from the chaotic response is a convex function with an unstable fixed point, which is an intersection of the map and the diagonal line (Fig. 5.4b(iv)). A sampled field potential in the vicinity of the unstable fixed point goes away from the fixed point, and then, is pushed back to the vicinity of the fixed point because of the convexity of the map. As it goes away again from the unstable fixed point, the sampled field potential wanders around on the one-dimensional map for ever. This one-dimensional map provides clear evidence for chaotic response of the hippocampal CA3 to mossy fiber stimulation. In other words, it is not the chaotic activity of individual pyramidal cells but chaotic mass activity of a pyramidal cell assembly in response to mossy fiber stimulation. This implies that activity of a neuronal assembly follows a deterministic dynamical rule, and this is an important point of view when we investigate dynamical features of the brain. In this sense, neural network models composed of units, namely synchronous neuronal assemblies, might be a good strategy of the brain research.

We can investigate how much phase-locked and chaotic field potential responses are spatially coherent in the hippocampal CA3 by obtaining mutual correlation functions between the field potentials recorded simultaneously at four different places as shown in Fig. 5.1. In both cases of the phase-locked and chaotic responses, mutual correlation functions have a large peak at the center (t = 0). This indicates that field potential responses to mossy fiber stimulation are well synchronized across the hippocampal CA3 along a sagittal axis (Refer to Hayashi and Ishizuka, 1995, for further details). However, magnifying those correlation functions, central peaks are slightly shifted from the central axis (t = 0). This indicates that a pyramidal cell assembly starts firing at a place in the hippocampal CA3, and then propagates to the surroundings. The propagation speed estimated from the shift of the central peak was 0.1 m/s.

Given that the propagation speed is 0.1 m/s, it takes less than 10 ms for synchronized burst firing of pyramidal cells to propagate from the proximal side to the distal side of the CA3 region along the sagittal axis. As the time being spent for the propagation is sufficiently short in comparison with the duration of epileptiform burst (about 40 ms) and the period of responses to a low-frequency input (about 100 ms), it might be allowed to consider that burst firings of pyramidal cells are almost synchronized across the hippocampal CA3 along the sagittal axis.

By the way, propagation of neuronal firing has been reported in longitudinal slices of the hippocampal CA3 [Miles et al., 1988]. Its propagation speed along the longitudinal axis is 0.15 m/s, and almost the same as that along the sagittal axis. This indicates that it takes more than 100 ms for neuronal firing to propagate from the septal side to the temporal side of the hippocampal CA3 along the longitudinal axis. The theta wave, which was recorded from the hippocampus of the rat running a track, takes almost the same time to propagate from the septal side to the temporal side of the hippocampal CA1 [Lubenov and Siapas, 2009]. Since the period of the theta rhythm is about 100 ms, the length of the hippocampus along the longitudinal axis is almost the same as the wavelength of the theta wave. In other words, neuronal activity of the hippocampus is not coherent along the longitudinal axis, and propagation of neuronal activity is not negligible.

Given that the activity of the hippocampal CA3 is coherent along the sagittal axis as mentioned above, the size of the synchronized assembly of pyramidal cells would be about one-tenth of the longitudinal length of the hippocampus. Such neuronal assembly might respond dynamically to inputs through mossy fibers in order to represent sensory information and/or generate new information. It is also conceivable that information received by the ventral hippocampus might be integrated with information in the dorsal hippocampus and the intermediate region between them by means of propagation of burst activity. If we averaged such activity across the hippocampus, it would be difficult not only to see dynamical features of neuronal assemblies but also to understand information processing mechanisms of the hippocampus.

References

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(Aug. 6, 2015)

## 6. Hippocampal CA3 Model: Rhythmic Activities and Chaotic Responses

It was shown in Chapter 5 that CA3 pyramidal cell assemblies in rat sagittal hippocampal CA3 slices cause phase-locked and chaotic responses to periodic stimulation of mossy fibers. It was also shown that burst activities propagate in both cases of the phase-locked and chaotic responses, although the burst activities are roughly synchronized along the sagittal axis of the hippocampal CA3. In the present chapter, we will look at complex spatiotemporal activities of a neural network model of the hippocampal CA3, which reproduces spontaneous rhythmic field activities and chaotic responses to periodic stimulation.

The present hippocampal CA3 model consists of 256 pyramidal cells placed on 16x16 lattice points and 25 inhibitory interneurons distributed across the network at regular intervals (Fig. 6.1). As the number of inhibitory interneurons is about 10% of the number of pyramidal cells in CA3 [Misgeld and Frotscher, 1986], 25 interneurons are included in the present model. This network model is a minimal model because a network model including less than 256 pyramidal cells is rather difficult to reproduce hippocampal rhythmic field activities.

Fig. 6.1   Hippocampal CA3 model. △: Pyramidal cells placed at 16x16 lattice points. : Inhibitory interneurons distributed across the network at regular intervals. Firing patterns of the pyramidal cell (▲) and the interneuron () are shown in Fig. 6.4 (ii) and (iii), respectively. Pyramidal cells are connected mutually to 8 surrounding pyramidal cells (for example, purple pyramidal cells) through excitatory connections. Each interneuron receives excitatory inputs from 16 surrounding pyramidal cells (for example, blue pyramidal cells), and inhibits the same pyramidal cells. Adjoining interneurons inhibit 4 common pyramidal cells. [adapted from Tateno et al., 1998]

Although 19- and 2-compartment models of the hippocampal CA3 pyramidal cell have been developed [Traub et al., 1991; Pinsky and Rinzel, 1994], the present pyramidal cell model is a single compartment model without dendrites [Tateno et al., 1998]. This conductance-based model well reproduces firing patterns and bifurcation phenomena (Fig. 6.2 b) observed in a rat hippocampal CA3 pyramidal cell (Fig. 6.2 a) (Refer to Tateno et al., 1998, for further details of the present pyramidal cell model).

A "low-threshold" Ca2+ current exists in the hippocampal CA3 pyramidal cell [Brown and Griffith, 1983]. This current is not included in the above 19- and 2-compartment models, but in the present single compartment model. The CA3 pyramidal cell is hyperpolarized after the cell fires because a Ca2+-dependent K+ current is activated. Then, the low-threshold Ca2+ current is activated when the membrane potential is returning to the resting potential. As activation of the low-threshold Ca2+ current depolarizes the pyramidal cell more than the threshold of the Na+ spike, the cell fires again. In other words, the low-threshold Ca2+ current allows the CA3 pyramidal cell to fire spontaneously. Moreover, as the Na+ spike triggers a “high-threshold” Ca2+ spike, which causes slow after depolarization, a burst of Na+ spikes occurs. Activated Ca2+ currents increases the Ca2+ concentration inside the cell, and a long-lasting hyperpolarization is produced by activation of the Ca2+-dependent K+ current because the Ca2+ concentration decreases slowly. The membrane potential returns gradually from the hyperpolarized state to the resting potential, and the low-threshold Ca2+ current is activated again. Consequently, the cell causes a spike burst repeatedly. Long interburst intervals are produced by the long-lasting hyperpolarization.

It can be considered that spike bursts are caused by the interaction between the fast oscillation produced by the Na+ and K+ currents and the slow oscillation produced by the Ca2+ and Ca2+-dependent K+ currents. This interaction depends on the amount of depolarization. Therefore, when the cell is depolarized by a dc current, burst intervals decrease with increase in the depolarization, and the firing pattern of the pyramidal cell bifurcates from burst firing to repetitive firing through chaotic firing (Fig. 6.2 a and b). The bifurcation diagram shows that the period 1 bifurcates to the chaos through a cascade of period doubling bifurcations with decrease in the dc current applied to the cell (Fig. 6.3).

Fig. 6.2   Spontaneous firing of the hippocampal CA3 pyramidal cell. (a) Intracellular recording from a rat hippocampal CA3 pyramidal cell ((i) period 1, (ii) chaos, and (iii) - (iv) periodic bursts). The dc current being applied to the soma is (i) 0.3, (ii) 0.2, (iii) 0.1, and (iv) 0 nA. (b) Firing of the hippocampal CA3 pyramidal cell model ((i) period 1, (ii) period 2, (iii) chaos, and (iv) - (vi) periodic bursts). The dc current being applied to the pyramidal cell model is (i) 0.4, (ii) 0.3, (iii) 0.27, (iv) 0.2, (v) 0.1, and (vi) 0 nA. [adapted from Tateno et al., 1998]

Fig. 6.3   Bifurcation diagram of the hippocampal CA3 pyramidal cell model. The abscissa is the dc current being applied to the cell model. The ordinate is the interspike interval. The period 1 bifurcates to the chaos through a cascade of period doubling bifurcations with decrease in the dc current, and furthermore, the chaos bifurcates to periodic burst firing. [adapted from Tateno et al., 1998]

The inhibitory interneuron model is a fast-spiking type containing only Na+, K+, and leak currents. Parameters of those currents have been adjusted to reproduce firing patterns of the hippocampal interneuron [Kawaguchi and Hama, 1987] observed in the experiments. Time constants of the excitatory and inhibitory synaptic currents have also been adjusted to reproduce excitatory and inhibitory synaptic potentials [Miles and Wang, 1986; Miles, 1990] observed in the experiments, respectively.

It has been well known that NMDA- and AMPA-receptor channels exist in excitatory synapses in the hippocampus. The Ca2+ current of NMDA-receptor channels, which triggers long-term potentiation of EPSPs caused by AMPA-receptor channels, plays an important role in inducing a plastic change in synaptic weight. On the other hand, it has been supposed that EPSPs caused by AMPA-receptor channels contribute to spread spike bursts of pyramidal cells to surrounding areas, and to enhance synchronization of spike bursts in a wide range of the neural network [Lee and Hablitz, 1989]. In this chapter, our hippocampal CA3 model contains only excitatory synapses of AMPA-receptor type, and plastic change in excitatory synaptic weight is disregarded. Spatiotemporal activity of a hippocampal CA3 model, whose excitatory synapses have a feature of spike-timing-dependent synaptic plasticity (STDP), will be shown in Chapter 7.

Each pyramidal cell is locally connected to 8 surrounding pyramidal cells through excitatory connections. Each inhibitory interneuron receives excitatory inputs from 16 surrounding pyramidal cells, and inhibits the same pyramidal cells. The region where a inhibitory interneuron inhibits overlaps partially with the region where an adjacent inhibitory interneuron inhibits. In other words, adjacent inhibitory interneurons inhibit 4 common pyramidal cells.

Since the present hippocampal CA3 model is composed by pyramidal cells firing spontaneously, spatiotemporal activity occurs based on spontaneous population activity of pyramidal cells. Although we can see firing of individual neurons in detail in numerical simulation using this model, it is quite difficult to record individual firings of many cells simultaneously in the experiments. It is therefore necessary to obtain the field potential from the spatiotemporal activity of the CA3 model in order to compare it with experimental results.

In the experiments shown in Chapter 5, spikes of each pyramidal cell was not recorded, but field potential was recorded using extracellular microelectrode placed on the pyramidal cell layer. This is because the field potential recording is a conventional method to observe activity of neuron population. Since field potential is a voltage drop produced in surrounding tissues and mediums by field current (mainly synaptic currents) associated with neuronal activity, spatial distribution of the field current density is required to obtain the field potential from activity of the CA3 model. However, it is difficult to see a precise distribution of the field current density. Given that the resistance of the environment does not vary with time, it can be assumed that the time course of the field current is proportional to that of the field potential. If so, we can compare the field current, which is obtained by summing up local synaptic currents, with the field potential recorded in experiments.

In the present study, the field current is defined by the sum of synaptic currents occurring in 16 pyramidal cells located at the center of the CA3 network model. As it is assumed that an electrode is placed at the center of the network, synaptic current occurring in a distant pyramidal cell is estimated smaller depending on the distance between the center and the pyramidal cell (refer to Tateno et al., 1998 and Hayashi, 2001 for further details). Although Na+ and K+ currents generating spikes are not included in this field current, high-frequency components are included in the field current because impulses are sent from surrounding pyramidal cells and interneurons. These high-frequency components were removed by a low-pass filter (50 Hz) in order to observe filed current rhythms in the frequency range below 50 Hz.

Fig. 6.4   Spontaneous spatiotemporal activity of the hippocampal CA3 network model. (a) Epileptic rhythm. Cpp (excitatory connection weight between pyramidal cells) = 0.008 μS. (b) δ rhythm. Cpp = 0.005 μS. (c) θ rhythm. Cpp = 0.003 μS. (d) β rhythm. Cpp = 0.001 μS. Excitatory and inhibitory connection weights between pyramidal cells and inhibitory interneurons are fixed. (i) Spike rasters. The ordinate is the number given to each neuron: top 256 and bottom 25 neurons on the ordinate are pyramidal cells and inhibitory interneurons, respectively. For example, top 16 pyramidal cells on the ordinate correspond to the 16 pyramidal cells in the top line of the network model. The abscissa is the time. (ii) Firing patterns of the pyramidal cell (▲ in Fig. 6.1). (iii) Firing patterns of the inhibitory interneuron ( in Fig. 6.1). (iv) Field currents. Note that vertical scales are different. [adapted from Tateno et al., 1998]

The field current shows four kinds of rhythmic activities depending on the excitatory connection weight between pyramidal cells (Fig. 6.4 (iv)). Fig. 6.4 (i) shows spike rasters. Dots indicate spikes of pyramidal cells. The abscissa is the time, and the ordinate is the number given to each neuron. Pyramidal cells, #1 and #16, are placed at the top left and the top right of the network, respectively, and pyramidal cell #256 is placed at the bottom right. The neurons from #266 to #281 are inhibitory interneurons. We can see spatiotemporal activity corresponding to each rhythmic activity. Firing patterns of the pyramidal cell (▲ in Fig. 6.1) and the inhibitory interneuron ( in Fig. 6.1) are shown in Fig. 6.4 (ii) and (iii), respectively.

When excitatory connections between pyramidal cells are weak, firings of pyramidal cells are hardly synchronized (Fig. 6.4 d(i)), and burst firing hardly occurs either (Fig. 6.4 d(ii)). This is because the recurrent inhibition is relatively strong. The high-threshold Ca2+ current is hardly activated, and consequently bursts of Na+ spikes hardly occur. Moreover, as the Ca2+ concentration inside the cell does not increase so much, hyperpolarization due to Ca2+-dependent K+ current is not so large, and a long silent period does not occur. Consequently, the amplitude of the field current oscillation is small, and its rhythmicity is not clear (Fig. 6.4 d(iv)). The power spectrum of the field current oscillation is broad, and small humps exist at 15 and 30 Hz (refer to Tateno et al., 1998, for further details). This field current oscillation was classified into the β rhythm.

Clear field current oscillation appears with increase in excitatory connection weights between pyramidal cells (Fig. 6.4 c(iv)). Since the power spectrum of the field current oscillation has a sharp peak at 6 Hz, this oscillation can be classified into the θ rhythm. Pyramidal cells receiving excitatory inputs through somewhat stronger connections cause clear spike bursts, and interburst intervals due to the Ca2+-dependent K+ current are also clear (Fig. 6.4 c(ii)). These interburst intervals correspond to the period of the field current oscillation at a theta frequency.

 Movie Fig. 6.5   Spontaneous spatiotemporal activity of the hippocampal CA3 model causing a θ rhythm. ●: Pyramidal cells. Squares, ■ and ■, represent the firing and resting states of the pyramidal cell, respectively. The time constant of transition from the excited state (■) to the resting state (■) is larger than the spike duration, and the animation is replayed slowly, in order to see propagation of spike bursts clearly. （Click Link to see the movie.）

Spike bursts do not synchronize across the network, but propagate in different directions (Fig. 6.4 c(i) and Fig. 6.5). As it had been supposed that the θ rhythm is coherent across the network, it was surprising that a rhythmic field current oscillation was observed in spite of irregular spatiotemporal activity. The reason would be as follows: when one extracellular electrode is used to record the field potential and a spike burst passes by the electrode repeatedly, a rhythmic field potential oscillation can be observed regardless of the direction of each propagation. The present results show at least that rhythmic activities can be reproduced even if spike bursts are not synchronized across the network. Moreover, slow increase and decrease of the amplitude of the field current oscillation, which resembles the waxing and waning of the θ rhythm observed in the rat brain, is also reproduced (Fig. 6.4 c(iv)). This slow change in the amplitude is attributable to complex propagation of excitatory waves. In other words, the field current oscillation often collapses because of collisions between spike bursts propagating in different directions.

Given that spike bursts are synchronized excessively across the neural network, the field potential shows a large amplitude oscillation like an epilepsy. However, the amplitude of the θ rhythm observed in the rat brain is not so large. Probably, spike bursts are synchronized in a relatively small region, and the synchronized burst activity propagates. Therefore, the network would not get into an excessively synchronized state, while maintaining a small amplitude rhythmic activity. Each propagating spike burst (excitatory wave) trails an inhibitory area produced by recurrent inhibition and deep hyperpolarization due to Ca2+-dependent K+ current. This inhibitory area could suppress the network to fall into an excessively excited state, and also contribute to adjust intervals between excitatory waves passing by the electrode to a theta period.

When excitatory connections between pyramidal cells are strengthened more, the rhythmic activity changes to a δ rhythm (Fig. 6.4 b(iv)), whose histogram has a peak at 3 Hz. The amplitude of the δ rhythm is larger than that of the θ rhythm, and fluctuates irregularly. The spike raster shows that neuronal assemblies of various sizes, in which spike bursts occur in synchrony, appear around the network (Fig. 6.4 b(i)). As excitatory connections are strengthened, spike bursts tend to be synchronized with each other. However, spike bursts are not synchronized completely across the network, and still propagate irregularly. The reason why the frequency of the δ rhythm is lower than that of the θ rhythm is that pyramidal cells cause burst firings intensely due to stronger excitatory inputs from surrounding pyramidal cells (Fig. 6.4 b(ii)). The intense burst firing increases the Ca2+ concentration inside the cell, and a deeper hyperpolarization is caused by the Ca2+-dependent K+ current. Consequently, the interburst interval corresponding to the period of the rhythmic field current increases.

When excitatory connections between pyramidal cells are strengthened fully, the network model causes epilepsy showing a large amplitude field current oscillation at a frequency of 2 Hz (Fig. 6.4 a(iv)). Although pyramidal cells repeat an intense spike burst (Fig. 6.4 a(ii)), those bursts are not synchronized across the network, but still propagate (Fig. 6.4 a(i)).

The delta (Fig. 6.4 b(iv)) and theta (Fig. 6.4 c(iv)) rhythms have an oscillatory aspect, though they are somewhat irregular. So, we expected that those oscillations were low-dimensional chaos. However, geometrical structure of the attractors reconstructed in a three-dimensional space was very complex, and it was difficult to get a sign of low-dimensional chaos (refer to Hayashi, 2001, for further details). This situation is similar to that it is difficult to provide certain evidence for chaos by analyzing EEG data. Probably, synchronization of neuronal activity is insufficient, and consequently dynamical dimensions of those rhythmic activities are not low sufficiently.

Fig. 6.6   Response of the spontaneous rhythmic activity of the hippocampal CA3 model to periodic stimulation. (a) Epilepsy, (b) δ rhythm, (c) θ rhythm, (d) β rhythm. The abscissa and the ordinate are the stimulus frequency (T is the interpulse interval) and the stimulus intensity, respectively. Stimulation was applied to all of the pyramidal cells simultaneously. : chaotic response. , , , and are 1:1, 1:2, 1:3, and 1:4 phase-lockings, respectively. [adapted from Tateno et al., 1998]

Response of rhythmic activity of the hippocampal CA3 model can be observed by applying periodic stimulation (Fig. 6.6) in a similar way to the experiments shown in Chapter 5. The epileptic and δ rhythms cause chaotic responses, and the δ rhythm is easier to cause 1:1 phase-locking than the epileptic rhythm (Fig. 6.6 a and b). The θ rhythm causes 1:1 phase-locking in a wide range of the phase diagram (Fig. 6.6 c). This indicates that 1:1 phase-locking occurs easily when the intensity of spike bursts reduces and synchronization of spike bursts is less sufficient. In other words, the network would be easier to respond to the stimulation when robustness of the rhythmic activity reduces. In the case of the β rhythm, rhythmic activity is not clear (Fig. 6.4 d(iv)), and its power spectrum is broad including high-frequency components [Tateno et al., 1998]. Consequently, the region of 1:1 phase-locking gets smaller in the phase diagram, and chaotic response does not occur (Fig. 6.6 d).

Figure 6.7 is an example of chaotic response of the δ rhythm. Chaotic response of the epileptic rhythm resembles the chaotic response of the δ rhythm. Although the spike raster shows pyramidal cells fire spike bursts in response to stimulus pulses, spike bursts are not the same (Fig. 6.7 (i)). Interburst intervals and the number of spikes within the burst fluctuate irregularly, and pyramidal cells often do not fire (Fig. 6.7 (ii)). Consequently, various sizes of pyramidal cell assemblies appear around the network, and the field current reflecting the number of fired neurons and the degree of synchronization fluctuates irregularly (Fig. 6.7 (iii)). One-dimensional map obtained from a sequence of field currents sampled at 60 ms after each stimulus pulse is a convex function, which shows features of low-dimensional chaos well (Fig. 6.7 (iv)).

Fig. 6.7   Chaotic response of the δ rhythm to periodic stimulation. A hippocampal CA3 model. (i) Spike raster. (ii) Firing pattern of the pyramidal cell ( in Fig. 6.1). (iii) Field current. Dots under the field current are current pulses for stimulation. (iv) One-dimensional map obtained from a sequence of field currents sampled at 60 ms after each stimulus pulse. [adapted from Tateno et al., 1998]

In the present model, all of the pyramidal cells were stimulated simultaneously by current pulses. Therefore, as shown in the spike raster (Fig. 6.7 (i)), spike bursts hardly propagate. It is supposed that if the CA3 network is stimulated locally imitating inputs to the hippocampal CA3 through mossy fibers, response of the local region propagates to surrounding areas. However, such propagation hardly occurs even if a local region of this CA3 model is stimulated. Spike-timing-dependent synaptic plasticity (STDP) of excitatory synapses is necessary to cause spike bursts propagating to surrounding areas. Such propagation of spike bursts will be shown in Chapter 7.

References

1. Brown DA and Griffith WH (1983) Persistent slow inward calcium current in voltage-clamped hippocampal neurons of the guinea-pig. Journal of Physiology 337: 303-320.
2. Hayashi H (2001) "Chaos in the brain." Shokabo, Tokyo [in Japanese].
3. Kawaguchi Y and Hama K (1987) Two subtypes of non-pyramidal cells in rat hippocampal formation identified by intracellular recording and HRP injection. Brain Research 411: 190-195.
4. Lee W-L and Hablitz JJ (1989) Involvement of non-NMDA receptors in picrotoxin-induced epileptiform activity in the hippocampus. Neuroscience Letters 107: 129-134.
5. Miles R (1990) Synaptic excitation of inhibitory cells by single CA3 hippocampal pyramidal cells of the guinea-pig in vitro. Journal of Physiology 428: 61-77.
6. Miles R and Wang RKS (1986) Excitatory synaptic interactions between CA3 neurons in the guinea-pig hippocampus. Journal of Physiology 373: 397-418.
7. Misgeld U and Frotscher M (1986) Postsynaptic-GABAergic inhibition of nonpyramidal neurons in the guinea-pig hippocampus. Neuroscience 19: 193-206.
8. Pinsky PF and Rinzel J (1994) Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons. Journal of Computational Neuroscience 1: 39-60.
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10. Traub RD, Wong RKS, Miles R, and Michelson H (1991) A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances. Journal of Neurophysiology 66: 635-650.
(Sep. 28, 2015)

## 7. Hippocampal CA3 Model: STDP and Propagation of Neuronal Activity

In the previous chapter, spontaneous rhythmic field activities were reproduced using a hippocampal CA3 network model, and it was shown that some of the rhythmic field activities caused chaotic responses to a periodic current pulse stimulation, besides phase-locked responses. It was also shown that neuronal activities were not synchronized across the network, but propagated irregularly on the network that produced a rhythmic field activity. However, since synaptic weights of the network model were fixed, new propagation did not start from a local site stimulated repeatedly. It would be necessary that local stimulation strengthens excitatory connections between pyramidal cells radially from the stimulus site to the surroundings in order to produce new propagation from the stimulus site. Such new propagation of neuronal activities caused by a radial pattern of strengthened excitatory connections could overcome recurrent inhibition and spontaneous neuronal activities.

The spike-timing-dependent synaptic plasticity (STDP) has been demonstrated in various regions of the brain [Levy and Steward, 1983; Magee and Johnston, 1997; Markram et al., 1997; Debanne et al., 1998; Bi and Poo, 1998; Nishiyama et al., 2000]. This type of synapse is strengthened a little bit when firing of a presynaptic cell precedes firing of a postsynaptic cell, and weakened a little bit when the firings occur in reverse order. Therefore, if firings of neurons caused by a local stimulation fired surrounding neurons, synapses having a feature of STDP would be strengthened in the radial direction from the stimulus site to the surroundings, and would be weakened in the opposite direction. Repeating the local stimulation, the area where excitatory connections are strengthened radially would spread from the stimulus site, and consequently firings could propagate a long way from the stimulus site to the surroundings.

Fig. 7.1   STDP function approximated by two exponential functions. The synaptic connection is strengthened (Ft) > 0) when a presynaptic spike precedes a postsynaptic spike (Δt < 0), and weakened (Ft) < 0) when the spikes occur in reverse order (Δt > 0). [adapted from Yoshida and Hayashi, 2004]

The hippocampal CA3 network model in the present chapter is the same as that in Chapter 6 except for a STDP feature of synapses. The STDP rule in Fig. 7.1 is a simple approximation of the experimental results obtained by Bi and Poo (1998), and is defined by two exponential functions as follows [Yoshida and Hayashi, 2004]:

$F(\Delta t)= \begin{cases} \ \ \ M \mbox{exp}(\Delta t/\tau) \qquad & \mbox{if} \ \ -T \le \Delta t <0\\ -M \mbox{exp}(\Delta t/\tau) \qquad & \mbox{if} \quad 0 < \Delta t \le T \qquad \qquad \qquad \mbox{(7.1)}\\ \qquad 0 & \mbox{otherwise} \end{cases}$

Δt denotes the relative spike timing between pre- and postsynaptic spikes, and is defined by the following equation:

Δt   = (the time when a presynaptic neuron fires a spike)
-   (the time when a postsynaptic neuron fires a spike)     (7.2)

Parameters, M, τ, and T, are 0.05, 20 ms, and 100 ms, respectively. Each excitatory connection weight between pyramidal cells, Cpp, is modified by the following equation:

$C_{\mbox{pp}} \rightarrow C_{\mbox{pp}} + C_{\mbox{max}}\ F(\Delta t) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mbox{(7.3)}$

Therefore, when a presynaptic spike precedes a postsynaptic spike, the connection weight Cpp increases because of positive Ft), and when the spikes occur in reverse order, the connection weight Cpp decreases because of negative Ft). However, the connection weight Cpp is limited to the range Cmin (= 0.0015 μS) ≤ CppCmax (= 0.005 μS). If Cpp > Cmax, Cpp is set to Cmax, and if Cpp < Cmin, Cpp is set to Cmin.

Now, let’s look at the spontaneous spatiotemporal activity of the hippocampal CA3 network model whose synapses are subject to the STDP rule. Spike timing between pre- and postsynaptic spikes fluctuates with time in the network causing complex spatiotemporal activity spontaneously. Therefore, strength of each excitatory connection having a feature of STDP also fluctuates with time.

Here, the vector $\scriptstyle{\vec{V}_{i}}$ is defined as follows [Yoshida and Hayashi, 2004]:

$\vec{V}_i = \sum_{j} \vec{v}_{ij}/C_{\mbox{max}}(1+\sqrt{2}) \qquad \qquad \qquad \qquad \qquad \qquad \mbox{(7.4)}$

The vector $\scriptstyle{\vec{v}_{ij}}$ points to a postsynaptic cell i from a presynaptic cell j, and the length of the vector is proportional to the connection weight Cpp between the cells. The Vector $\scriptstyle{\vec{V}_i}$, which is proportional to the summed vector $\scriptscriptstyle{\sum_{j} \vec{v}_{ij}}$, represents the spatial asymmetry of the strength of excitatory connections to the postsynaptic cell i. In other words, the excitatory connections are more strengthened in the direction of the vector $\scriptstyle{\vec{V}_i}$. $\scriptstyle{C_{\mathrm{max}}(1+\sqrt{2})}$ is a factor to normalize the summed vector $\scriptscriptstyle{\sum_{j} \vec{v}_{ij}}$. Therefore, activity of the pyramidal cells is most likely to propagate to the direction of the vector $\scriptstyle{\vec{V}_i}$. If weights of excitatory connections to the postsynaptic cell are spatially symmetric, $\scriptstyle{\vec{V}_i \; = \; 0}$.

The CA3 network model, whose initial excitatory connection weights Cpps are the same (0.0033 μS), causes complex spatiotemporal activity and its field current shows a theta rhythm as shown in Chapter 6 (Fig. 6.5). As the connection weights are the same in the initial state, initial strength of connections to each postsynaptic pyramidal cell is spatially symmetric ($\scriptstyle{\vec{V}_i}$ = 0) except for pyramidal cells on the edge of the network. When excitatory connection weights Cpps are modified through the STDP rule shown in Fig. 7.1, some synaptic connections are strengthened and the other synaptic connections are weakened by complex spatiotemporal activity. In other words, vectors $\scriptstyle{\vec{V}_i}$ grow from pyramidal cells in different directions as shown in Fig. 7.2 a and Fig. 7.3 (movie). Consequently, excitatory connection weights from surrounding presynaptic pyramidal cells to a postsynaptic pyramidal cell become asymmetric spatially, and moreover, the spatial distribution of the vectors $\scriptstyle{\vec{V}_i}$ becomes non-uniform over the whole network.

When synaptic connection weights Cpps, which are subject to the STDP rule, are modified by the complex spatiotemporal activity, the connection weights are split into two groups, maximum and minimum, and there are few intermediate connection weights (Fig. 7.2 b). Moreover, Cpps converge to a split distribution regardless of initial values of Cpps. This is because the STDP function in Fig. 7.1 is symmetric with respect to the origin (0, 0), and the present CA3 network model is a recurrent network, in which pyramidal cells are connected to each other. In other words, when the connection from the pyramidal cell j to the pyramidal cell i is strengthened depending on the spike timing, the connection from the cell i to the cell j is weakened, and consequently the average of connection weights Cpps converges to 0.0033 μS (Fig. 7.2 c). Although connections are modified with time by complex spatiotemporal activity, strengthened and weakened connection weights are always balanced, and the averaged connection weight is invariable. Thus, spontaneous rhythmic field activity is controlled and maintained by STDP, and the frequency of the theta rhythm is maintained at about 7 Hz (Fig. 7.2 d).

Complexity of the spatiotemporal activity of the CA3 network model (Fig. 6.5 movie), in which synaptic connection weights are fixed, is almost the same as that of the CA3 network model (Fig. 7.3 movie), in which synaptic connection weights are modified through the STDP rule shown in Fig. 7.1. Moreover, field current rhythms (theta rhythms) are almost the same in both network models (Fig. 7.2 e). These suggest that modification of synaptic weights through the STDP rule does not have a large influence on spatiotemporal activity and field current rhythm of the network.

Fig. 7.2   Spontaneous activity of a hippocampal CA3 network model. Excitatory connection weights between pyramidal cells are subject to the STDP rule shown in Fig. 7.1. (a) Spatial distribution of vectors $\scriptscriptstyle{\vec{V}_i}$ (i = 1-256) 40 sec after the initial state. Cpps are 0.0033 μS initially. : Pyramidal cell. The bar from each pyramidal cell i represents the $\scriptscriptstyle{\vec{V}_i}$, and indicates the direction in which excitatory connections are strengthened. (b) Distribution of excitatory connection weights Cpps 40 sec after the initial state. Initial values of Cpps are (i) 0.0025, (ii) 0.0033, and (iii) 0.0040 μS. Excitatory connection weights are split into two groups with time. (c) Average of Cpps across the network. Initial values of Cpps are 0.005, 0.004, 0.0033, 0.0025, or 0.0015 μS, and Cpps are subject to the STDP rule from 20 sec onward. The average of Cpps converges to 0.0033 μS regardless of the initial values. (d) Average of frequencies of field current oscillations observed at five different locations in the network. Standard deviations are also indicated. Three groups indicated by 10, 40, and 70 sec show frequencies obtained from field current oscillations at different periods of time (3-20, 33-50, and 63-80 sec), respectively. Three bars in each group have different initial Cpp values (left: 0.0025, center: 0.0033, and right: 0.0040 μS). Excitatory connection weights Cpps are subject to the STDP rule from 20 sec onward. Frequencies of field current oscillations converge to about 7 Hz regardless of the initial conditions. (e) Field current oscillations (theta rhythms) obtained from spatiotemporal activity of the network. It is assumed that a recording electrode is placed at the center of the network. (i) All of the connection weights Cpps are the same and fixed (0.0033 μS). (ii) Connection weights Cpps are subject to the STDP rule. Each connection weight Cpp is therefore modified with time by activities of surrounding pyramidal cells. [adapted from Yoshida and Hayashi, 2004]

 Movie Fig. 7.3   Spontaneous spatiotemporal activity of the hippocampal CA3 network model, in which excitatory connection weights between pyramidal cells are modified through the STDP rule shown in Fig. 7.1. ●: Pyramidal cell. ■: Firing of a pyramidal cell. ■: Resting state of a pyramidal cell. The time constant of transition from ■ to ■ is larger than the spike width and the movie is replayed slower in order to see propagation of excitatory waves easily. The yellow bar extending from each pyramidal cell i (i = 1-256) is the vector $\scriptscriptstyle{\vec{V}_i}$. We can see that the vectors grow in various directions, because excitatory connection weights between pyramidal cells Cpps are not uniformly modified with time by spontaneous complex activity of pyramidal cells. (Click here to see the movie.)

Now, let’s look at responses of the hippocampal CA3 network model to local stimulation. The network model causes complex propagation of excitatory waves spontaneously, and the frequency of the field rhythmic activity obtained from the complex spatiotemporal activity is in the theta range (about 7 Hz). A local area (for example, four pyramidal cells inside the square in Fig. 7.4) of the hippocampal CA3 network model is stimulated by a burst of current pulses. When the frequency of the burst stimulation (i.e. the reciprocal of the interburst interval) is a little bit higher than that of the theta rhythm, excitatory connections Cpps are strengthened in radial directions from the stimulus site, and are weakened in opposite directions (Fig. 7.4 b). Consequently, an excitatory wave propagates repeatedly in the radial direction like a ripple spreading from the stimulus site (Fig. 7.5, movie). This repetitive radial propagation continues for a while after termination of stimulation, and then fades away. When the frequency of the burst stimulation is a little bit lower than that of the theta rhythm, excitatory connections are not strengthened in radial directions from the stimulus site (Fig. 7.4 c), and repetitive radial propagation of an excitatory wave does not occur.

Fig. 7.4   Radial patterns of excitatory connection weights Cpps strengthened by local stimulation. A hippocampal CA3 network model, in which excitatory connection weights Cpps are modified through the STDP rule shown in Fig. 7.1. Spontaneous rhythmic field activity is a theta rhythm (about 7 Hz). (a) Spatial distribution of $\scriptscriptstyle{\vec{V}_i}$ (i = 1-256) before applying the local stimulation. This panel is the same as Fig. 7.2 a. (b) Spatial distribution of $\scriptscriptstyle{\vec{V}_i}$ 160 sec after the beginning of a local burst stimulation (8Hz). The stimulation is applied simultaneously to four pyramidal cells inside the square. Excitatory connections Cpps are strengthened in radial directions from the stimulus site. The number of pulses within a burst for stimulation is 3, interpulse intervals are 10 ms, and interburst intervals are 125 ms. (c) Spatial distribution of $\scriptscriptstyle{\vec{V}_i}$ 160 sec after the beginning of a local burst stimulation (5 Hz). The local stimulation fails to strengthen excitatory connections Cpps in radial directions from the stimulus site. The number of pulses within a burst for stimulation is 3, interpulse intervals are 10 ms, and interburst intervals are 200 ms. [adapted from Yoshida and Hayashi, 2004]

 Movie Fig. 7.5   Radial propagation of excitatory wave produced by local stimulation. A hippocampal CA3 network model, in which excitatory connections between pyramidal cells are modified through the STDP rule shown in Fig. 7.1. Spontaneous rhythmic field activity is a theta rhythm (about 7 Hz). Four pyramidal cells inside the square shown in Fig. 7.4 b are stimulated simultaneously by a burst stimulation (8 Hz). Excitatory waves caused by the local stimulation propagate radially from the stimulus site. ●: Pyramidal cell. ■: Firing of a pyramidal cell. ■: Resting state of a pyramidal cell. The time constant of transition from ■ to ■ is larger than the spike width and the movie is replayed slower in order to see propagation of excitatory waves easily. The yellow bar extending from each pyramidal cell i (i = 1-256) is the vector $\scriptscriptstyle{\vec{V}_i}$. The local stimulation is terminated 9 sec after the beginning of the movie. Note that radial propagation of an excitatory wave occurs repeatedly after the termination of the local stimulation. (Click here to see the movie)

Here, Drad, which indicates how much excitatory connections are strengthened in radial directions from the stimulus site, is defined as follows:

$D_{rad}=\frac{\sum_{i=1}^n (\vec{V}_i \centerdot \vec{I}_i)}{n} \qquad \qquad \qquad \qquad \qquad \qquad \mbox{(7.5)}$

$\scriptstyle{\vec{V}_i}$ is the vector defined by Eq. 7.4. $\scriptstyle{\vec{I}_i}$ is the unit vector whose direction is from the center of the stimulus site to the location of the pyramidal cell i. n is the number of pyramidal cells inside a circle, whose center coincides with the center of the stimulus site as shown in Fig. 7.6. The radius of the circle is about six times larger than the distance between the nearest pyramidal cells. Summation is done among pyramidal cells inside the circle. When the direction of the vector $\scriptstyle{\vec{V}_i}$ is identical with the direction of the vector $\scriptstyle{\vec{I}_i}$ at all pyramidal cells inside the circle, the value of Drad is the largest, and all of the directions of vectors $\scriptstyle{\vec{V}_i}$ are exactly radial. The value of Drad decreases with increase in the difference between $\scriptstyle{\vec{V}_i}$ and $\scriptstyle{\vec{I}_i}$, and a small value of Drad indicates that directions of vectors $\scriptstyle{\vec{V}_i}$ are not necessarily radial. Therefore, radial propagation of excitatory waves is clear when the value of Drad is large, and the radial propagation fades away with decrease in the value of Drad.

Fig. 7.6   Estimation of Drad, which indicates how much excitatory connections are strengthened in radial directions from the stimulus site (refer to Eq. 7.5 for definition of Drad). $\scriptscriptstyle{\vec{V}_i}$ is the vector defined by Eq. 7.4. $\scriptscriptstyle{\vec{I}_i}$ is the unit vector whose direction is from the center of the stimulus site to the location of the pyramidal cell i. A circle with the center coinciding with the center of the stimulus site is used to estimate Drad. [adapted from Yoshida and Hayashi, 2004]

The average and the standard deviation of Drads obtained by stimulating nine different locations of the hippocampal CA3 network model are shown in Fig. 7.7. Initial conditions are the same in those simulations. When a local area of the network model is stimulated by a 8 Hz burst stimulation, Drad increases rapidly (Fig. 7.7 a) indicating that Cpps are strengthened in radial directions from the stimulus site. Then, an excitatory wave starts propagating repeatedly along the radial pattern of strengthened Cpps. On the other hand, when the network model is stimulated at a 5 Hz burst stimulation, Drad decreases slowly and reaches zero (Fig. 7.7 a). Therefore, no radial pattern of strengthened Cpps is developed by the stimulation, and radial propagation does not occur either.

When the 8 Hz burst stimulation is terminated, the averaged Drad decreases gradually, as shown in Fig. 7.7 b. Although repetitive radial propagation of excitatory wave fades away with decrease in the averaged Drad, it continues for a while after termination of the local stimulation. This is because the radial pattern of enhanced Cpps produced by the local stimulation is maintained for a while. Possibly, we may consider that this maintained propagation is a short-term memory state of the network.

Excitatory connection weight Cpp averaged across the network is always constant regardless of occurrence of radial propagation of excitatory waves (Fig. 7.7 c). Even if some excitatory connections are strengthened to develop a spatial pattern of enhanced connections, some other excitatory connections are weakened, and consequently the averaged connection weight Cpp is invariable. This suggests that STDP may contribute to homeostasis of the excitation level across the network. In other words, STDP seems to be a safety device controlling excitation level of neural networks so that an extreme excitatory state is not produced by storing memory.

Since rich recurrent excitatory connections exist in the hippocampal CA3 region [Li et al., 1994], it has been supposed that the hippocampus is an associative memory system storing memories with plastic change in recurrent connection weights [ Marr, 1971; McNaughton and Morris, 1987; Treves and Rolls, 1994 ]. Recently, it has been demonstrated that synapses of those recurrent connections cause plastic change through STDP [ Debanne et al., 1998; Bi and Poo, 1998 ], and it has also been supposed that STDP is a mechanism to store sequence information in recurrent networks within the framework of the associative memory. On the other hand, it has been well known that the hippocampal CA3 region frequently causes spontaneous activity [ Buzsáki, 2002; Strata, 1998; Wu et al., 2002 ]. These suggest that strength of recurrent connection weights being subject to a STDP rule changes gradually due to spontaneous spiking of pyramidal cells in the CA3 region. Consequently, associative memory patterns stored in a recurrent network may be damaged gradually by the spontaneous activity. Moreover, mechanisms that distinguish neuronal activities of local areas containing memory patterns from spontaneous background activity still remain unknown.

In this chapter, it was demonstrated that an excitatory wave propagated repeatedly from a stimulus site using a recurrent network model with STDP synapses. These excitatory waves are caused on a spatial pattern of excitatory connection weights organized by local stimulation, and are supposed to be a neuronal activity distinguished from background spontaneous activities. It is also supposed that plastic change in connection weights due to STDP is rather a mechanism to cause organized propagation of excitatory waves (theta waves) than a mechanism to store sequence information in a recurrent network. Theta waves observed in the hippocampal CA1 of the rat running on a linear track really propagate from the septal side to the temporal side of the hippocampus [ Lubenov and Siapas, 2009 ], and it is supposed that one of the causes is the projection of propagating theta waves in the hippocampal CA3. If that is the case, it would be natural that propagation of the theta rhythm in CA3 is caused by inputs from the entorhinal cortex (EC) and the dentate gyrus (DG) carrying spatial information. Moreover, it is attractive to suppose that sequence information transmitted to CA1 from EC is coded at phases within a cycle of the CA1 traveling theta wave projected from CA3. However, the coding mechanisms are not yet fully understood at present.

If the hippocampal CA3 neural network model shown in this chapter were made slender, it would be possible to cause traveling excitatory waves along the long axis by stimulating one of the short sides, like a traveling theta waves observed by Lubenov and Siapas (2009) . However, unidirectional propagation of the excitatory wave caused by local stimulation is not clear. It seems to be necessary to introduce some factors, for example an anisotropic distribution of inhibitory connections, into the network (Refer to Samura and Hayashi, 2012 , for further details).

References

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(June 26, 2016)

## 8. Chaotic Responses of the Rat Primary Somatosensory Cortex

Excitatory wave propagation as a collective neuronal activity was mentioned using hippocampal CA3 models in Chapters 6 and 7 after chaotic response of the hippocampal CA3 slice was demonstrated in Chapter 5. Now, let’s go back to the talk on chaotic activities of the brain.

In Chapter 5, chaotic response of the hippocampal CA3 region to mossy fiber stimulation was demonstrated using hippocampal slices. As we used hippocampal slices, we could access activities of the CA3 network simultaneously at different locations using several glass microelectrodes. However, it is supposed that the slices, which are cut off from the brain and maintained in an artificial environment, are not so natural as the hippocampus existing in the brain. In this chapter, chaotic responses of the somatosensory cortex of the brain will be demonstrated using anesthetized rats.

Afferent fibers from peripheral sensory receptors pass the medulla oblongata, and then form a bundle called the medial lemniscus (ML). Those fibers project to the thalamus, and sensory signals from periphery are transmitted to the somatosensory cortex through thalamo-cortical neurons. As afferent fibers from sensory receptors run in a bundle in ML, imitation sensory signals can be delivered to the somatosensory cortex through the thalamus by stimulating ML. The number of afferent fibers excited by ML stimulation increases with increase in the stimulus intensity. In other words, the number of impulses transmitted simultaneously to thalamo-cortical neurons can be controlled by the stimulus intensity. Excitation of many fibers in ML by a strong ML stimulation corresponds to simultaneous excitation of many sensory receptors in periphery.

When peripheral sensory receptors are stimulated, impulse trains are generated in peripheral nerve fibers. Therefore, if a periodic pulse stimulation is used to stimulate ML fibers, it is possible to investigate the nature of responses of the somatosensory cortex to the stimulation imitating sensory signals. Field potential responses, whose high-frequency components were removed by a low-pass filter (50 Hz, 24 dB/oct), were analyzed to show dynamical features of the somatosensory cortex responses including chaos.

Since the field potential response of the primary somatosensory cortex depends on the depth of anesthesia, the depth of anesthesia should be maintained at a constant level during experiments. Here, two criteria are introduced to judge the depth of anesthesia: (1) spindle oscillations (about 10 Hz) are caused by a single shock to ML fibers (Fig. 8.1 a), and (2) spontaneous field potential rhythm has a dominant frequency in the frequency range of the delta rhythm (Fig. 8.1 b).

Fig. 8.1   Spindle oscillations and the delta rhythm recorded from the surface of the rat primary somatosensory cortex. (a) Spindle oscillations caused by a single shock to ML fibers. Each trace is the average of 20 records. The frequency of the spindle oscillation is about 10 Hz. Amplitudes of the current pulses for ML stimulation are 0.11 (upper), 0.15 (middle), and 0.25 (lower) mA. (b) Power spectrum of spontaneous field potential rhythm. ML fibers were not stimulated. Five power spectra were averaged. Each power spectrum was obtained using a field potential record for 30 sec. The frequency fo at the peak of the spectrum is in the frequency range of the delta rhythm. [adapted from Ishizuka and Hayashi, 1996]

The field potential response of the somatosensory cortex to a single current pulse for ML stimulation consists of a small positive potential with a short latency and a subsequent large negative potential (Fig. 8.2). It has been supposed that the positive response originates in activities of excitatory synapses in the deep layer of the somatosensory cortex caused by an input from the thalamus, and the negative field potential reflects polysynaptic activities in the superficial layer of the somatosensory cortex following the synaptic activities in the deep layer [Morin and Steriade, 1981; Sasaki et al., 1970]. Although the average amplitude of the negative field potential increases with increase in the amplitude of the current pulse for ML stimulation (Fig. 8.2), the amplitude of the field potential fluctuates when ML fibers are stimulated repeatedly by a single current pulse. Therefore, synaptic activities in the superficial layer of the primary somatosensory cortex caused by a periodic input from the thalamus are quite complex.

Fig. 8.2   Field potential responses of the rat primary somatosensory cortex to single current pulses for ML stimulation. The ML stimulation causes a small positive field potential with a short latency, and subsequently, a large negative field potential. Twenty field potentials caused by single current pulses with the same amplitude were averaged. The amplitude of the averaged negative field potential increases with increase in the amplitude of the single current pulse for ML stimulation. The amplitudes of the current pulses are 0.11, 0.13, 0.15, 0.2, 0.25, and 0.3 mA. Spindle oscillations shown in Fig. 8.1 a occur with a long delay after these field potential responses. [adapted from Ishizuka and Hayashi, 1996]

Spindle oscillations (Fig. 8.1 a) are caused in the thalamo-cortical circuit by a single shock of ML fibers after the fast field potential responses of the somatosensory cortex (Fig. 8.2). This means that repetitive inputs from ML to the thalamus are interfered by the spindle oscillations in the thalamo-cortical system. In other words, when spindle oscillations occur, thalamo-cortical relay neurons in the thalamus are inhibited by thalamic reticular neurons, and responsiveness of the thalamo-cortical neurons to synaptic inputs from the periphery is low [Glenn and Steriade, 1982]. Therefore, it is hard for sensory signals arising from the periphery to reach the cortex through the thalamus.

However, EPSPs generated in thalamo-cortical neurons by repetitive ML stimulation are integrated gradually, and overcome IPSPs caused by thalamic reticular neurons. Consequently, thalamo-cortical neurons are depolarized, spindle oscillations disappear, and periodic signals from ML are transmitted efficiently to the primary somatosensory cortex [Jahnsen and Llinás, 1984; McCormik and Pape, 1990; Lopes da Silva, 1991]. This functional behavior of the thalamus is called the gating function. In other words, the role of the thalamus changes from a spindle generator to a relay station. Such periodic signals transmitted to the somatosensory cortex through the thalamus interact with the delta rhythm of the somatosensory cortex. Phase-locked and chaotic field potential responses of the somatosensory cortex are caused mainly by the interaction between the periodic afferent signal and the delta rhythm.

Fig. 8.3   Phase diagram of the field potential response of the rat primary somatosensory cortex to periodic ML fiber stimulation. I and Ith are the amplitude of the current pulse for ML stimulation and the threshold amplitude of the current pulse causing a negative field potential response, respectively. fi and fo are the frequency of ML stimulation (i.e. the reciprocal of the interpulse interval) and the frequency of the delta rhythm, respectively. In the regions having symbols (, , , , and ), 1:1, 1:2, 1:3, 2:2, and 2:4 phase-locked responses occur, respectively, and in the regions having symbols (, , , and ), chaotic responses occur. [adapted from Ishizuka and Hayashi, 1996]

The phase diagram of the field potential response of the somatosensory cortex to periodic ML stimulation is shown in Fig. 8.3. Parameters are the normalized intensity I/Ith and the normalized frequency fi/fo of the periodic ML stimulation. The amplitude of the current pulse I for ML stimulation was normalized by the threshold amplitude of the current pulse Ith causing a negative field potential response, and the stimulus frequency f was normalized by the frequency of the delta rhythm fo occurring spontaneously in the somatosensory cortex. Since the frequency fo varied from one experiment to the next, the frequency fo used for normalizing the stimulus frequency f was determined by obtaining a power spectrum using a 20 ms segment of the time series of the spontaneous field potential just before the beginning of stimulation. In the regions having symbols (, , , , and ), 1:1, 1:2, 1:3, 2:2, and 2:4 phase-lockings occur, respectively, and in the regions having symbols (, , , and ), chaotic responses occur. Each response indicated by is a typical chaotic response whose one-dimensional map is a convex function with an unstable fixed point, as shown in Fig. 8.4 ciii. Although each response in the intermediate regions (, , and ) between regions of phase-lockings looks like a mixture of two kinds of phase-lockings, one-dimensional maps obtained from those responses demonstrate that they have chaotic features (see Ishizuka and Hayashi, 1996 for details).

Fig. 8.4   Phase-locked and chaotic responses of the rat primary somatosensory cortex to periodic ML stimulation. (a) 1:1 phase-locking. fi/fo = 1.43. I/Ith = 1.87. (b) 1:2 phase-locking. fi/fo = 2.86. I/Ith = 1.87. (c) Chaos observed in the region having symbols in Fig. 8.3. fi/fo = 3.37. I/Ith = 3.18. (i) Upper and lower traces show field potential responses and current pulses for ML stimulation, respectively. (ii) Attractors reconstructed in the two-dimensional space (V, dV/dt). (iii) One-dimensional maps. Each map was obtained from a sequential series of field potentials sampled at Ts (= 14 ms) after each stimulus current pulse. [adapted from Ishizuka and Hayashi, 1996]

In the case of 1:1 phase-locking, a field potential response occurs every period of ML fiber stimulation (Fig. 8.4 ai), and trajectories produce a closed loop in the phase space (V, dV/dt) (Fig. 8.4 aii). The one-dimensional map was obtained from a sequential series of field potentials sampled at Ts (= 14 ms) after each stimulus current pulse. It shows a single cluster on the diagonal line (Fig. 8.4 aiii), and demonstrates that this response pattern is the 1:1 phase-locking. In the case of 1:2 phase-locking, a large field potential response occurs every two periods of ML fiber stimulation (Fig. 8.4 bi). Trajectories produce a large loop with a small one (Fig. 8.4 bii). Since field potentials sampled every period of the ML stimulation (Ts = 14 ms) make a sequence of alternate large and small field potentials, the one-dimensional map obtained from the sequence shows two clusters being symmetric with respect to the diagonal line (Fig. 8.4 biii).

In the case of the chaos observed in the region having symbols in Fig. 8.3, field potential responses to a periodic ML stimulation are irregular (Fig. 8.4 ci), and irregular trajectories reconstruct a strange attractor in the two-dimensional phase space (V, dV/dt) (Fig. 8.4 cii). The one-dimensional map obtained from a sequential series of field potentials sampled every period of ML stimulation (Ts = 14 ms) is a convex function with an unstable fixed point (Fig. 8.4 ciii). This map clearly demonstrates that the sequence of irregular responses is chaotic. Moreover, a stroboscopic cross-section of the attractor can be obtained by plotting field potentials sampled every period of ML stimulation on the two-dimensional phase space (V, dV/dt). Dependence of the cross-section on Ts clearly shows stretching (Fig. 8.5 a-d) and folding (Fig. 8.5 f-i) of the attractor. This stretching and folding of the attractor is also a reliable evidence for chaos.

Fig. 8.5   Cross-sections of the strange attractor reconstructed from the chaotic response observed in the region having symbols in Fig. 8.3. Each cross-section was reproduced by plotting field potentials, which were sampled at Ts after each current pulse for ML stimulation, on the two-dimensional phase space (V, dV/dt). fi/fo = 3.37, and I/Ith = 3.18. Dotted lines show the contour of the strange attractor shown in Fig. 8.4 cii. (a-i) Ts = 8, 9, 11, 14, 17, 20, 23, 24, 26 ms, respectively. The cross-section is stretched (a-d), and then folded (f-i). (j) Fifty field potential responses to ML stimulation are superposed. The stimulus current pulses are superposed at the same position on the panel. [adapted from Ishizuka and Hayashi, 1996]

Since a single electrode touching the surface of the somatosensory cortex was used to record the field potential, it was rather difficult to get information about the spatial size of synchronized neuronal activity in the superficial layer of the somatosensory cortex. However, as the amplitude of the field potential recorded from the surface of the cortex is comparable to that recorded from hippocampal CA3 slices, it is supposed that neurons in a relatively small neuronal assembly around the electrode respond in synchrony to the ML stimulation.

As mentioned above, chaotic responses can be observed clearly even in the brain of the anesthetized rat. This means that the deterministic nonlinear dynamics can be applied in a wide range of dynamical behavior of the brain.

References

1. Glenn LL and Steriade M (1982) Discharge rate and excitability of cortically projecting intralaminar thalamic neurons during waking and sleep states. J. Neurosci. 2: 1387-1404.
2. Ishizuka S and Hayashi H (1996) Chaotic and phase-locked responses of the somatosensory cortex to a periodic medial lemniscus stimulation in the anesthetized rat. Brain Res. 723: 46-60.
3. Jahnsen H and Llinás R (1984) Electrophysiological properties of guinea-pig thalamic neurons: an in vitro study. J. Physiol. 349: 205-226.
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(Aug. 4, 2016)

## 9. Concluding Remarks

After a chaotic response of the Onchidium giant neuron to periodic stimulation was found [Hayashi et al., 1982], chaotic responses of the Onchidium pacemaker neuron [Hayashi et al., 1983, 1985, 1986] and the squid giant axon [Matsumoto et al., 1984] to periodic stimulation were demonstrated in succession, and the study of neurochaos made rapid progress. Then, it has also been demonstrated that the Onchidium pacemaker neuron causes spontaneous chaotic activity without periodic stimulation [Hayashi and Ishizuka, 1992], besides single neuron models [e.g. Chay, 1985; Kaas-Petersen, 1987; Canavier et al., 1990]. Moreover, these studies have demonstrated not only chaotic features of single neurons but also bifurcations between periodic and chaotic firings depending on the stimulus intensity, the stimulus frequency, the membrane potential level, and so on. Fruitful results of these studies contributed greatly to understanding diverse and complex activities of single neurons within the framework of deterministic nonlinear dynamics.

It was expected that complex activities of the brain might also be understood within the framework of deterministic nonlinear dynamics after chaotic activities of single neurons were demonstrate. Since neurons interact with each other through synaptic connections in the brain, activities of individual neurons tend to be collected into activities of neuronal assemblies. It is therefore important to investigate synchronized activities of neuronal assemblies in order to understand dynamical features of the brain activity. Even so, a brain state, in which neuronal activities are synchronized in a wide range of the brain, is epileptic and rather unusual. Since the amplitude of rhythmic EEGs connected closely with brain functions is generally much smaller than that of epileptic EEGs, it is supposed that neuronal activities are synchronized in small regions of the brain. In other words, synchronized neuronal activities in a normal brain state are mixed together over a wide range of the brain, and the degrees of freedom of the total brain activity would not be low enough to see features of a low dimensional chaos. Actually, it is quite difficult to get evidence for chaos by analyzing human EEGs recorded from the scalp. It would be a good way to investigate activities of a local neuronal assembly in the brain. One of the proper means to record activities of such a small neuronal assembly is to record local field potential using an extracellular micro electrode.

As mentioned in Chapter 5, the CA3 region of sagittal hippocampal slices causes spontaneous field potential oscillations reflecting synchronized burst firings. The descending phase of the field potential corresponds to collective hyperpolarization of pyramidal cells, and its bottom phase, namely the period between bursts, acts like a relative refractory period because spikes are hard to be fired. Therefore, the number of pyramidal cells fired by periodic synaptic inputs decreases at the hyperpolarized state, and the amplitude and latency of the action potential of each pyramidal cell fluctuate. Consequently, the field potential response of the CA3 region to periodic mossy fiber stimulation fluctuates irregularly. This irregular field potential responses show chaotic features clearly [Hayashi and Ishizuka, 1995]. Moreover, as mentioned in Chapter 8, chaotic field potential responses of the somatosensory cortex to periodic ML stimulation have been demonstrated clearly using anesthetized rats [Ishizuka and Hayashi, 1996].

Relationship between brain functions and chaotic neuronal activities has not been understood well. However, finding of chaotic activities of single neurons and the brain has clarified that not only predictable periodic activities but also unpredictable complex activities can be understood within the framework of deterministic nonlinear dynamics. This contributed greatly to progressing studies on dynamical features of single neurons and the brain. From a different point of view, it would be possible in future to investigate information processing mechanisms of the brain using a brain model, which consists of a huge number of neurons or neuronal assemblies represented by deterministic nonlinear dynamical equations.

It should be noted that synchronized burst activities of small neuronal assemblies propagate across a recurrent network such as the hippocampal CA3. It is supposed that a small neuronal assembly causes phase-locked and chaotic responses to local synaptic inputs, and then those responses propagate to surrounding areas. Propagation of activities of small neuronal assemblies might distribute information throughout the network, and might avoid falling into a widespread synchronized state such as an epilepsy.

Propagation of burst firings has been observed not only in sagittal slices of the hippocampal CA3 [Hayashi and Ishizuka, 1995], but also in longitudinal slices of the CA3 [Miles et al., 1988], as mentioned in Chapter 5. It has also been reported that rhythmic activities in the hippocampal CA3 disappear when AMPA receptor channels are blocked [Wu et al., 2002], suggesting that excitatory recurrent connections are vital for rhythm generation. Moreover, complex propagation of excitatory waves occurs spontaneously in a hippocampal CA3 network model [Tateno et al., 1998] as mentioned in Chapter 6, and organized excitatory waves propagate from a local stimulus site when excitatory recurrent connection weights are modified through a STDP rule [Yoshida and Hayashi, 2004] as shown in Chapter 7. In those days when above studies were carried out, it was supposed that recurrent networks like the hippocampal CA3 were suitable for associative memory, and propagation of excitatory waves in those networks did not receive much attention. However, it has been demonstrated by Lubenov and Siapas (2009) that theta waves propagate along the longitudinal axis of the hippocampal CA1 of the rat running on a linear track. Theta waves propagate actually, though it had been supposed that theta rhythms are synchronized across the network. These traveling theta waves have been confirmed by Patel et al. (2012). However, it seems to be rather difficult for the hippocampal CA1 to cause traveling theta waves because CA1 has few excitatory recurrent connections. It has been supposed that one of the causes of the traveling theta waves in CA1 is the projection of traveling theta waves in CA3 onto CA1.

If the hippocampal CA3 is an associative memory system, the CA3 causing spontaneous excitatory waves is not suitable for the memory function. Synaptic connection weights are modified by the spike propagation, and consequently memory patterns embedded in the network would be lost gradually, because recurrent connection weights have a feature of spike-timing-dependent plasticity. It would be better to suppose that the hippocampal CA3 causing excitatory waves is a rhythm generator rather than an associative memory. In other words, spike-timing-dependent synaptic plasticity might be a mechanism to organize traveling theta waves rather than to embed information in a recurrent network as a memory pattern. Sequence learning of places, which is crucial for animal’s navigation, is closely connected to the hippocampal theta rhythm [e.g. O’Keefe and Recce, 1993]. It would be an important future issue to solve the problem of how sequence information is coded in the hippocampus based on traveling theta waves rather than a spatially synchronized theta rhythm.

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(Aug. 21, 2016)