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5Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 1, pp. 19-34. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220901
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 37N25, 92B20, 37G99
Excitation of a Group of Two Hindmarsh - Rose Neurons with a Neuron-Generated Signal
I. R. Garashchuk, D. I. Sinelshchikov
We study a model of three Hindmarsh-Rose neurons with directional electrical connections. We consider two fully-connected neurons that form a slave group which receives the signal from the master neuron via a directional coupling. We control the excitability of the neurons by setting the constant external currents. We study the possibility of excitation of the slave system in the stable resting state by the signal coming from the master neuron, to make it fire spikes/bursts tonically. We vary the coupling strength between the master and the slave systems as another control parameter. We calculate the borderlines of excitation by different types of signal in the control parameter space. We establish which of the resulting dynamical regimes are chaotic. We also demonstrate the possibility of excitation by a single burst or a spike in areas of control parameters, where the slave system is bistable. We calculate the borderlines of excitation by a single period of the excitatory signal.
Keywords: chaos, neuronal excitability, Hindmarsh-Rose model
1. Introduction
The ability of a neuron to fire a spike or a burst in response to a synaptic current plays crucial role in the functioning of biological neural networks [1]. The reaction of a neuron to different patterns of input current defines its computational properties. Hodgkin defined three classes of neuronal excitability based on the measured frequency-current curves in response to electric stimulation of a squid axon [2]. As shown later, different behavior of different neurons
Received August 04, 2022 Accepted September 01, 2022
Sections 3, 4 of the work were supported by RFBR grant 20-31-90122. Sections 5, 6 were supported by RSF grant 19-71-10048.
Ivan R. Garashchuk [email protected] Dmitry I. Sinelshchikov [email protected]
HSE University
ul. Tallinskaya 34, Moscow, 123458 Russia
is related to different bifurcation mechanisms of excitability [3]. Thus, the more modern classification separates neurons into categories of resonators and integrators, based on the bifurcation mechanisms involved [1]. Excitability of the integrator neurons occurs in the vicinity of the saddle-node bifurcation of the stable equilibrium, lying either on the invariant circle, for monostable integrators, or outside of a spiking limit cycle, for bistable integrators. On the other hand, resonator neurons are excitable because they are close to the Andronov-Hopf bifurcation of the resting state, either subcritical or supercritical. This difference in the excitability mechanism leads to different behavior of these types of neurons. Integrators fire based on the integral of the input current, while resonators react more to the frequency of the input current, see [1] for details.
Dynamical systems are well known to be an excellent tool for describing biological neuronal systems [1]. Although the optimal model of choice depends on a particular problem at hand [1,4], in general a reliable model should be capable of reproducing the main types of neuronal firing patterns of the corresponding neurons, such as tonic spiking, bursting and resting [5, 6]. The Hodgkin -Huxley theory is the conductance-based formalism often used to derive dynamical systems describing excitable cells, such as neurons [7]. Multistability also plays an important role in such biophysical processes. Thus, some authors use modified Hodgkin-Huxley type models to incorporate certain biological mechanisms into the models [8-10]. In this work, we use the Hindmarsh - Rose model, also derived within the Hodgkin-Huxley formalism, to simulate the neuronal activity [11]. It is a biologically plausible model capable of capturing typical behavior of neurons.
There have been a number of works dedicated to studying models of biological neurons, neuronal networks and the Hindmarsh-Rose model in particular (see, e.g., [12-21] and references therein). Dynamics and firing patterns of two coupled Hindmarsh - Rose neurons can be significantly different from those of one neuron [22]. We are interested in excitation of the group of two neurons in a stable quiescent state by different firing patterns typical for biological neurons. Thus, we consider the excitatory signal to be also generated by a similar neuron. The signal is transmitted via a directional connection to one of the two neurons in the slave group.
The model and the numerical methods that are used are described in Section 2. In Section 3 we briefly discuss properties of the master and slave systems. In Sections 4 and 5 we study the excitability of the stable resting state of the slave system in different areas of the control parameters. In Section 6 we investigate the excitation of the slave system by a single period of the external signal from the master system, leading to tonic spiking or bursting in the slave system in the areas of control parameters, where the slave system is bistable. In the last section we briefly summarize and discuss our results.
2. The main system of equations
We use the Hindmarsh-Rose model to simulate each individual neuron. The couplings between the neurons are simulated as direct electrical connections between their membranes. Such connections between the neurons are described by linear couplings in equations [23]. We suppose that the second and the third neurons have a bidirectional coupling, i. e., they form a fully-connected group and the first neuron acts as the master system, generating different types of signals. They are transmitted to the slave system, namely, the group of two fully-connected neurons, via the one-directional connection with the second neuron, see Fig. 1. We also consider that neurons have multicompartment dendrites, which allows us to introduce constant currents
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Fig. 1. Scheme of the neuronal network being simulated
applied to each neuron. These constant currents, I1, I2 = I and I3 = I, control the excitability of each neuron. Omitting the void couplings, the governing system of equations for the neuronal system in Fig. 1 is
where xi, i = 1, 2, 3 are the potentials on the membranes of the respective neurons, while yi and zi, i = 1, 2, 3 are the fast and slow ionic currents, respectively. We assume that the neurons are identical and, hence, we use the same values of parameters in all equations of (2.1). The system of equations (2.1) is a slow-fast one, because the time scales of zi variables are defined by the small parameter r = 0.0021. The other parameters have the following fixed values: a = 1, b = 3, c = 1, d = 5, s = 4 and x° = —| [14]. In the slave system we have chosen the coupling strengths between the neurons to be equal: D23 = D32 = 0.1, the same as in [22]. The strength of the unidirectional connection D12 is one of the control parameters. We assume that the values of currents applied to both neurons in the slave system are constant and the same I2 = I3 = I, i.e., we suppose that these currents come from the same source, as in Fig. 1. We use the value of I as a control parameter as well.
For numerical integration of the model (2.1) we use a quasi-constant step size method of the Klopfenstein-Shampine family [24, 25], which is suitable for stiff systems. To calculate the spectrum of Lyapunov exponents, we use the standard algorithm by Bennetin et al. [26], after skipping the transient process.
3. Properties of the master and the slave systems
We begin with a brief discussion of the dynamics in the slave system in the absence of any driving signal (D12 = 0). In this case, system (2.1) possesses the symmetry of swapping the
X1 = y1 — ax3 + bx2 — z1 + I1, i)i = c — dxf — y1, Z1 = r(s(x1 — x0) — z1),
x 2 = i2 — ax2 + bx2 — z2 + I2 + D32 (x3 — x2) + D12 (x1 — x2), i2 = c — dx2 — i2,
z>2 = r(s(x2 — x0) — z2), x 3 = i3 — ax3 + bx3 — z3 + I3 + D23 (x2 — x3 ^ y3 = c — dx3 — Уз,
z3 = r(s(x3 — x0) — z3^
(2.1)
neurons, namely, x3 o x2, y3 o y2, z3 o z2. However, since only the second neuron is connected to the first one, this symmetry is broken when D12 = 0. Now we briefly describe possible types of dynamics in this subsystem, following [22]. In the area of low external currents, I < 1.2895, there exists a stable equilibrium point for which x2 = x3, y2 = y3, z2 = z3. At I = 1.2895 it undergoes the subcritical Andronov-Hopf bifurcation and becomes unstable. Let us denote this value of current as IAH = 1.2895. This fixed point is a stable node for I < 0.685 and a stable focus for 0.685 < I < I ah . Consequently, the slave system has some properties of a resonator because of this subcritical Andronov-Hopf bifurcation.
Notice that for I > 1.2760 there also exists an asynchronous attractor (with windows of periodicity), corresponding to chaotic bursting modes [22], making the slave system bistable in the interval 1.2760 < I < I ah . In this bistability region it is possible to force the slave system to transit from the stable equilibrium over the separatrix into the basin of another attractor with an input signal of a finite length. If it happens, the slave system will converge to this attractor if the input signal is disabled, and will continue to burst tonically. For I < 1.2760, after the crisis of the chaotic attractor, the stable equilibrium becomes the global attractor. We assume that the mechanism of the attractor crisis is the following: while crossing the subcritical Andronov-Hopf bifurcation point in the direction of decreasing current, the equilibrium becomes stable, and a saddle limit cycle emerges. The stable manifold of the limit cycle plays the role of the separatrix, separating the basins of attraction of the stable equilibrium and the chaotic attractor. As I decreases further, this manifold eventually intersects with the chaotic attractor, which leads to attractor crisis. After that, the chaotic set in the phase space is not attractive.
(a) (b)
Fig. 2. Regular spiking regimes of a single neuron at (a) I1 = 3.50, = 33.56, (b) I1 = 5.70, T( = 8.10
The signal received by the slave system from the master neuron is more natural for neuronal systems than a sinusoidal or another artificial signal. The master system can exhibit all the main types of nontrivial firing patterns: periodic spiking, periodic bursting and chaotic bursting. However, the spiking regimes typically have quite short periods, see, e.g., signals at I1 = 5.70 and I1 = 3.50, presented in Figs. 2a, 2b, which have periods Tf = 8.10 and T% = 33.56, respectively. Such periodic spiking regimes are typical for one Hindmarsh-Rose neuron [14]. On the other hand, the natural periods of small damped oscillations in the neighborhood of the stable focus in the slave system at low currents are considerably lower than the periods of these periodic spiking regimes, see Fig. 3.
Contrary to the spiking patterns, periodic bursting regimes can have much longer periods, which are comparable to the periods of subthreshold oscillations near the stable fixed point. For example, the periodic bursting regime of the master neuron at I1 = 3.20 has 12 spikes per burst and an interburst period of T0 = 318.48 (see Fig. 4a). We use the Fourier transform to examine
450
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Fig. 3. Dependence of the period of small subthreshold oscillations of the slave system in the vicinity of the stable fixed point at low currents
J
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Fig. 4. Time series of the regular bursting signals of the master neuron at (a) I1 = 3.20, (b) I1 = 2.00, (c) I1 = 1.40
the characteristics of this signal. Intervals between the spikes within a regular burst consecutively increase with each spike. Therefore, there is no single period describing them. Thus, several smaller spectral maxima correspond to average intervals between spikes, the largest spectral components corresponding to T = 15.15, 11.36, 10.26. Bursting patterns with less spikes per burst are presented in Figs. 4b, 4c. The regime at I1 = 2.0 has 5 spikes per burst and period Tb = = 252.53, and the one at I1 = 1.4 has 3 spikes per burst and period T2 = 316.46. Given the ratios between the periods of the subthreshold oscillations close to the stable equilibrium in the low currents area and the periods of the spiking and bursting signals, we assume that the bursting signals fit better as the excitatory signal in this interval of currents.
4. Excitation of the slave system at low values of the constant external currents
Starting from the value of I where the stationary point in the slave system becomes a stable focus, we study how it reacts to excitation by a periodic bursting regime. We investigate the dependence of the slave system's response on two control parameters: the external current I and the coupling strength Dl2. We study whether the group of two neurons driven by the excitatory signal coming from the first neuron can fire bursts or spike tonically.
We present two-dimensional regions in the parameter space I, D12 that correspond to different types of dynamics of the slave system in Figs. 5, 6. In Fig. 5 we show at which parameters the slave neurons can consistently generate bursts or spikes, while in Fig. 6 we demonstrate regions of regular and chaotic dynamics.
Fig. 5. Borderlines of excitation of the subsystem of two neurons in a stable state for 0.680 < I < 1.285, 0 < D12 < 1. The black line BN2 shows the borderline of excitation of the second neuron, and the red line indicates the borderline of excitation of both neurons. In zone (I) both neurons burst tonically, in zone (II) none of the neurons fire bursts or spikes after the transient process, while in zone (III) only one neuron of the two generates bursts tonically, while the other one experiences subthreshold oscillations
Fig. 6. Borderline of chaos 0.680 < I < 1.285, 0 < D12 < 1. The filled area corresponds to chaotic motion
The following situations are possible: (I) the signal excites both neurons, and they fire bursts tonically; (II) both neurons exhibit subthreshold oscillations; (III) the signal makes only the second neuron fire consistently, while the third one exhibits subthreshold oscillations. The resulting regimes in case (I) can be both chaotic or periodic. In zone (II) we observed only regular subthreshold oscillations. The hybrid regimes in case (III) can be both regular and chaotic.
The borderline between chaotic and regular dynamics in Fig. 6 is quite complicated. The bottom line, which separates regular and chaotic motions in Fig. 6, and the black line in Fig. 5, which separates zone (II) from the others, coincide. There is a stiff transition from regular to chaotic dynamics when crossing that line.
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Fig. 7. Time series Xi, x3 for periodic regimes: (a) subthreshold oscillations in zone (II) at I — 1.0, D12 — 0.10, (b) regular bursting with period 3Tq in zone (II) at I — 1.13, D12 — 0.98
An example of regular subthreshold oscillations from zone (II) for I — 1.0, D12 — 0.10 is presented in Fig. 7a. This regime has the same period as the excitatory signal T — T^ — 322.59, capturing its frequency, which is common for subthreshold oscillations. Periodic regimes in zone (I) can be observed at relatively high coupling strengths. We present an example for I —
— 1.13, D12 — 0.98 in Fig. 7b. Despite its visual complexity, the largest Lyapunov exponent (LLE) of this regime is A — 0, confirming that it is a regular regime. Fourier analysis suggests that it has period T — 961.55 — 3T0b, i. e., the pattern of the slave system repeats itself once every three bursts of the master neuron, which can be observed in Fig. 7b. Overall, the periods of the regular regimes are multiples of T^ and a particular multiplier depends on the control parameters. For example, periodic bursting at I — 0.95, D12 — 0.90 has period T — 636.95 — 2Tq .
A chaotic bursting regime excited by the periodic bursting signal can be observed at I —
— 1.25, D12 — 0.5 (see Fig. 8a). It has the LLE A — 0.0117. Let us also remark that at the larger values of I, close to the point of the Andronov-Hopf bifurcation, especially in the bistability area, relatively small coupling strength is enough for the slave system to transition to tonic bursting. The lowest coupling strength sufficient for the slave system to transition from subthreshold oscillations to tonic bursting we have observed in this area of the control parameters with this partiular excitatory signal is D12 & 0.02. An example of a chaotic bursting regime with a relatively weak coupling strength at I — 1.285, D12 — 0.03 with LLE A — 0.0038 is shown in Fig. 8b. Although the two neurons of the slave system fire not with every burst of the excitatory signal, sometimes only one of the two neurons firing, nevertheless the slave system never fully transitions to subthreshold oscillations.
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Fig. 8. Time series xi, x^, x3 of chaotic bursting patterns in zone (I): (a) with strong coupling at I — 1.25, D12 — 0.5 and (b) with weak coupling at D12 — 0.03, I — 1.285
In zone (III) only the second neuron fires bursts consistently, while the third one experiences subthreshold oscillations. An example of periodic regime from this zone at I — 0.74, D12 — 0.75 is shown in Fig. 9a. A chaotic regime can be observed at I — 0.75, D12 — 0.6 (see Fig. 9b). It has LLE A — 0.00244.
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Fig. 9. Time series xi, x2, x3 of different regimes in zone (III): (a) regular regime at D12 — 0.75, I — 0.74, (b) chaotic regime at D12 — 0.6, I — 0.75
Considering that chaotic regimes were found only in zones (I) and (III), while in zone (II) we have only observed regular subthreshold oscillations, we assume that the chaotic set in the phase space, corresponding to chaotic bursts of the slave system, is responsible for emergence of chaos in the system.
The lowest coupling strength D12 necessary for transition of the slave system at a certain current I from subthreshold oscillations to tonic bursting differs between different bursting patterns of the master neuron. To illustrate this point, we use the signal generated by the master neuron at I1 = 2.00 with an interburst period of T\ = 252.53 and 5 spikes per burst (see Fig. 4b). Close to the Andronov-Hopf bifrucation at IAH, periods of the small damped oscillations in the
neighborhood of the stable equilibrium in the slave system are closer to Tb rather than T0 (see Fig. 3). The lowest pieces of the curves, separating tonic bursting from subthreshold oscillations in the slave system, excited by different bursting signals from the master neuron, are presented in Fig. 10. Longer bursts generated by the master neuron at I1 = 3.2 require smaller coupling strengths to excite the slave system at lower values of I. On the other hand, for higher values of I closer to I ah , the signal generated by the master neuron at I1 = 2.0 excites tonic bursting at lower values of the coupling strength. In this case, the lowest coupling strength leading to sustained bursting is D12 = 0.004.
I
Fig. 10. Borderlines of excitation of the subsystem of two neurons in a stable resting state for 1.15 < I < IAH, 0 < D12 < 0.12. The black line and the red line show, respectively, the borderline of excitation of the slave system (both neurons) by the signal generated by the master neuron at I1 = 2.0 and by the signal generated by the master neuron at I1 =3.20
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Fig. 11. Transient process, when the master neuron is spiking tonically at I1 = 3.50, and I = 1.285,
D12 =0.1
We cannot find sustained spiking or bursting response from the slave system, when the master neuron is in a periodic tonic spiking regime, instead of bursting (see examples of tonic
spiking in Fig. 2). That is, the whole region in Fig. 5 would correspond to subthreshold oscillations for the different spiking regimes we have used. We have only observed several spikes or bursts in the transient process, but then the system converges to subthreshold oscillations, and does not fire spikes or bursts anymore (see the example for I1 = 3.5, I = 1.285, D12 = 0.1 in Fig. 11). After the transient process presented in Fig. 11, the second neuron captures the frequency of the signal from the master, T2 = 33.56 in this case, while the third one oscillates with a longer period of T3 = 256.41.
5. Excitation of the slave system at high values of the constant external currents
Fig. 12. Borderlines of excitation of the slave system in the stable equilibrium for 5.398 < I < 6.198, 0 < D12 < 0.12. The black line and the red line show, respectively, the borderline of excitation by the spiking signal from the master neuron at I1 = 5.7 and by the bursting signal at I1 = 3.20
In this section we discuss excitation of the stable fixed point in the slave system at higher values of the constant current: 5.398 < I < 6.198. The value I = 6.1976 corresponds to the subcritical Andronov - Hopf bifurcation, and the fixed point is unstable for higher currents. While at I = 5.3978 the equilbrium undergoes the supercritical Andronov - Hopf bifurcation, becoming unstable at lower currents. We use two different types of signals from the master neuron to excite the slave system in the stable equilibrium state: bursting at I1 = 3.20 (see Fig. 4) and tonic spiking at I1 = 5.70 with a period of Tf = 8.10 (see Fig. 2b).
In this entire interval of currents the slave system is bistable: the stable equilibrium coexists with a quasi-periodic attractor [22]. The mean interspike intervals of the quasi-periodic regime vary with I from Tq1 & 9.05 at I = 5.40 to Tq & 6.85 at I = 6.20. Such values are of the same order as the interspike interval of the master neuron at I1 = 5.70.
In this area of the control parameters both bursting and spiking signals from the master neuron are viable for excitation of the slave system (see Fig. 12). This shows contrast with Section 4, where the spiking regimes were unable to make the slave system to spike/burst tonically. In addition, we have observed here no regimes with a single neuron in the slave system firing tonically, while the other one oscillates below the threshold, unlike in Section 4.
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Fig. 13. Excitation by a bursting signal from the master neuron at I1 = 3.2 with coupling strength D12 = = 0.03 and constant current applied to the slave system I = 5.5. (a) Transient process, (b) tonic spiking after the transient process
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Fig. 14. Excitation by spiking signal from the master neuron at I1 = 5.7 with coupling strength D12 = = 0.03 and constant current applied to the slave system I = 5.5. (a) Transient process, (b) tonic spiking after the transient process
If the excitation takes place, the resulting regimes have more in common with the quasi-periodic oscillations in the slave system than the excitatory signal. If the bursting signal is applied, the slave system spikes tonically after the transient process (see the example at D12 = = 0.03, I = 5.5 in Figs. 13a, 13b). The mean interspike interval only slightly differs from the mean interspike interval of the quasi-periodc oscillations in the slave system at I = 5.5: if- =
= 0.9903. On the other hand, it drastically differs from the interburst interval or the mean intraburst interspike interval of the excitatory bursting signal. Note that after the slave system starts spiking tonically, the spikes are fired even when the master neuron is in the quiescent periods between the bursts (see Fig. 13b). Thus, the external signal plays a key role in the process of excitation, making the slave system to transition from the equilibrium over the separatrix to tonic spiking, but its influence is relatively small after the transient process, and can be considered a small perturbation to the quasi-periodic regime in the slave system. In case the master neuron fires spikes with external current I1 = 5.7, choosing D12 = 0.03, I = 5.5 to be the same as before,
the difference between the mean interspike intervals of the resulting regime and the quasi-periodic
att.ractor becomes larger than in the case of bursting, S- = 0.9338 (see Fig. 14).
6. Excitation by a single burst or a single spike
i
Fig. 15. Borderlines of excitation of the slave system in the stable quiescient state for 1.2760 < I < IAH, 0 < D12 < 0.18. The red line shows the borderline of excitation by a single burst from the master neuron at I1 = 3.20 with 12 spikes per burst, the black line, by a single burst with 5 spikes from the master neuron I1 = 2.0, the green line, by a single burst with 3 spikes from the master neuron I1 = 1.4, and the blue line, by a single spike from the master neuron I1 = 3.5
In the areas of the control parameters, where the slave system is bistable, it is possible to force it to transition from the stable equilibrium, corresponding to quiescent neurons, over the separatrix into the basin of attraction to another regime, corresponding to tonic spiking or bursting, by a signal of a finite length. Then, even if the external signal is switched off, the slave system will continue to fire spikes/bursts tonically. In this section we explore the area 1.2760 < I < IAH with varying D12. We consider four types of excitatory signals: bursting regimes with different burst lengts and the number of spikes per burst, namely, the pattern with 12 spikes per burst at I1 = 3.2, the regime with 5 spikes per burst I1 = 2.0, the one with 3 spikes per burst at I1 = 1.4 and the spiking regime at I1 = 3.5. We disable the input signal coming to the second neuron after a single period of the excitatory signal to see if the slave system converges back to the stable equilibrium or to another attractor after the transient process.
The borderlines of excitation by a single period of different input signals in the parameter space (I1, D12) are presented in Fig. 15. In this case, the borderline of excitation for the excitatory spike has a much simpler shape than for the excitatory bursts. Examples of excitation of the slave system at I = 1.284 by different types of excitatory signal are presented in Figs. 16a-16d. If the excitation is possible, the resulting regime after the transient process will be chaotic or periodic, depending exclusively on the properties of the slave system, i.e., whether or not the particular value of I falls into a window of periodicity. Also notice that it is possible to excite the slave system by a single spike if the signal is disabled after that. On the other hand, if the
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Fig. 16. Excitation of the slave system at I = 1.284 by a single period of the exictatory signal: (a) D12 = = 0.1, I1 = 3.2, (b) D12 = 0.1, I1 = 2.0, (c) D12 = 0.1, I1 = 1.4, (d) D12 = 0.2, I1 = 3.5
periodic tonic spiking signal coming from the master neuron at I1 = 3.5 continued indefinitely, the slave system would eventually converge to subthreshold oscillations (see the end of Section 4).
Fig. 17. Rebound spike of the slave system at I = 5.6 in response to a brief hyperpolarizing pulse characteristic for the quescient periods of the master neuron between the spikes at I1 = 3.5 with D12 = 0.1
Regarding the interval of bistability with currents 5.398 < I < 6.198, the situation is more complicated. The slave system has properties of a resonator in this interval of I too. But there is also a significant difference between the membrane potential of the neurons of the slave system in the state of rest and the membrane potential of the master neuron in the quescient periods
between the spikes or bursts. These factors lead to occurence of rebound spikes in the slave system, which are followed by the slave system converging to the quasi-periodic attractor and spiking tonically after the signal from the master is disabled (see the example in Fig. 17). It is hard to determine which part of the input signal has more impact on the excitation of the slave system — the shape of the quescient period, promoting a rebound spike, or the spike or burst itself. Thus, a thorough study of excitability in this case should be carried out in a separate work.
7. Concluding remarks
We have studied a system of three neurons with directional electrical couplings, in which two fully-connected neurons that form the slave system receive an excitatory signal from the master neuron via a directional coupling. We used a biologically plausible model to simulate the dynamics of the neurons. We have shown that the slave group can be excited by a signal coming from the master neuron, to spike or burst tonically. We used different types of firing patterns of the master neuron as the excitatory signals. We have established areas of control parameters, in which the excitation takes place, for these types of signals.
In the areas of low external currents applied to the slave group, only bursting signals are able to excite the slave system to spike or burst tonically. Among the spiking signals we have used, we have not observed tonic spiking or bursting in the slave system. A small number of bursts could be observed in the transient process, but after that the slave system converged to subthreshold oscillations. In the case of bursting signals, we have observed three types of behavior of the slave system: subthreshold oscillations of both neurons, tonic bursting of both neurons and excitation of only one neuron within the slave group, while another one exhibits subthreshold oscillations. If both neurons exhibit subthreshold oscillations, we have observed only regular dynamics. In case at least one of them bursts tonically, chaotic or periodic behavior can be observed, depending on the particular values of the control parameters. We have observed stiff transition from regular subthreshold oscillations of both neurons in the slave system to chaotic tonic bursting with increasing coupling strength between the master and the slave systems.
We have also studied another area of control parameters with higher external currents. There the slave system is bistable: a quasi-periodic regime coexists with a stable equilibrium. Here, both spiking and bursting signals coming from the master neuron were viable for exciting the slave system to spike tonically. We have demonstrated the borderlines of excitation in the parameter space. In general, the resulting spiking patterns had more in common with the quasi-periodic regime of the slave system than the excitatory signal.
In the area of the control parameter space, where the slave system is bistable, we have also studied the possibility of excitation of the slave system by a single period of the excitatory signal, disabling the connection afterwards. There is an interval of bistability in the area of low external currents, where a chaotic attractor coexists with the stable equilibrium. We have calculated the borderlines of excitation by a single burst or spike in this area of the control parameters. It is interesting to notice that sometimes a single spike can excite the slave system to converge to the chaotic attractor and burst tonically. On the other hand, the infinitely long periodic spiking signal received by the slave system eventually led it to subthreshold oscillations. In another area of bistability with higher values of the external currents, the rebound spikes, leading to quasi-periodic tonic spiking in the slave system, play a very important role in the excitation process.
Thus, a careful study of excitability by a signal of a finite length in this region of parameters should be carried elsewhere.
Acknowledgments
The authors are grateful to anonymous reviewers for their valuable remarks.
Conflict of interest
The authors declare that they have no conflict of interest.
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