Bursting regimes in map-based neuron models coupled through fast threshold modulation

Borja Ibarz, Hongjun Cao, and Miguel A. F. Sanjuán

Phys. Rev. E 77, 051918 – Published 30 May 2008

Abstract

A system consisting of two map-based neurons coupled through reciprocal excitatory or inhibitory chemical synapses is discussed. After a brief explanation of the basic mechanism behind generation and synchronization of bursts, parameter space is explored to determine less obvious but biologically meaningful regimes and effects. Among them, we show how excitatory synapses without any delays may induce antiphase synchronization; that a synapse may change its character from excitatory to inhibitory and vice versa by changing its conductance, without any change in reversal potential; and that small variations in the synaptic threshold may result in drastic changes in the synchronization of spikes within bursts. Finally we show how the synchronization effects found in the two-neuron system carry over to larger networks.

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Received 7 January 2008

DOI:https://doi.org/10.1103/PhysRevE.77.051918

Authors & Affiliations

Borja Ibarz^1,*, Hongjun Cao^1,2, and Miguel A. F. Sanjuán¹

¹Nonlinear Dynamics and Chaos Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
²Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, People’s Republic of China

^*borja.ibarz@urjc.es

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Vol. 77, Iss. 5 — May 2008

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Figure 1
Return map $F_{α} (x, γ)$ for the fast variable of (a) the chaotic Rulkov model, with $γ = - 3$ and $α = 4.15$ , and (b) the nonchaotic Rulkov model, with $γ = - 4$ and $α = 6.0$ . With these values of $α$ both models are bursters, and the values of $γ$ have been chosen in the region of bistability. $S$ and $U$ denote stable and unstable fixed points, respectively, of the fast map. The shape of the fast maps is the only difference between the two variants of the model, with the consequence that in (a) we get irregular bursts, while in (b) bursts are regular.Reuse & Permissions
Figure 2
Nullcline diagrams of the Rulkov models. (a) Chaotic model with $α = 4.15$ , $σ = - 1.5$ . $N_{s}$ and $N_{u}$ are stable and unstable branches of the fast nullcline; $y = γ_{s n}$ marks the saddle-node bifurcation of the fast subsystem. The dashed horizontal line is the slow nullcline at $x = σ$ . The dotted, nearly horizontal line $Ξ_{min}$ represents the second iterate of the critical point $x = 0$ . $γ_{c r}$ is the value of $y$ of the external crisis bifurcation. $M_{spikes}$ is the manifold of average values of spiking in the fast subsystem at each fixed value of $y$ . (b) Nonchaotic model with $α = 6.0$ , $σ = - 1.2$ . Same labels as before; now $Ξ_{min}$ is at the reset level $x = - 1$ and $γ_{h}$ marks the homoclinic bifurcation.Reuse & Permissions
Figure 3
(Color online) Bursting orbit in phase space of (a) the chaotic Rulkov model, with $α = 4.15$ , $σ = - 1.2$ , and $η = 0.001$ , and (b) the nonchaotic Rulkov model, with $α = 6.0$ , $σ = - 1$ , and $η = 0.002$ . $N_{s}$ and $N_{u}$ denote the stable and unstable branches of fixed points of the fast subsystem, respectively. $Ξ_{max}$ and $Ξ_{min}$ are the maximum and minimum possible iterates of the fast map for each value of $y$ . $γ_{c r}$ , $γ_{h}$ , and $γ_{s n}$ mark the $y$ values of the external crisis, homoclinic, and saddle-node bifurcations, respectively.Reuse & Permissions
Figure 4
Two-parameter bifurcation diagram of the single neuron fast subsystem $x_{n + 1} = F_{α} (x_{n}, γ)$ of (a) the chaotic rulkov model and (b) the nonchaotic Rulkov model. The thick solid line $Γ_{s n}$ represents the curve of saddle-node bifurcations. The thick dashed lines $Γ_{c r}$ and $Γ_{h}$ represent the curves of external crisis and homoclinic bifurcations, respectively. In the shadowed region, bistability exists in the fast variable, and the neuron bursts if $σ$ has an appropriate value.Reuse & Permissions
Figure 5
Time evolution of two coupled neurons, with $g_{c} = 0.1$ , $σ = - 1.25$ , and $θ = 0$ in all cases. (a) Chaotic Rulkov neurons with $ν = 1$ (excitatory). (b) Chaotic Rulkov neurons with $ν = - 2$ (inhibitory). (c) Nonchaotic Rulkov neurons with $ν = 1$ (excitatory). (d) Nonchaotic Rulkov neurons with $ν = - 2$ (inhibitory).Reuse & Permissions
Figure 6
Simple characterizations of the synchronized dynamics of two chaotic Rulkov neurons. (a) Correlation dimension of the attractor as coupling strength increases from $g_{c} = 0$ , for excitatory $(ν = 0.0)$ and inhibitory $(ν = - 2.0)$ synapses. (b)–(d) Projection of the attractors on the $(x_{n, 1}, x_{n, 2})$ plane for three different synaptic reversal potentials. Dashed lines mark the synaptic threshold $(θ = - 1.4)$ . Maximal conductance is $g_{c} = 0.1$ in the three cases.Reuse & Permissions
Figure 7
(a) Dependence of Lyapunov exponents around the attractor as a function of coupling of two chaotic Rulkov neurons. The emergence of a synchronized state as $g_{c}$ increases is not reflected in the Lyapunov exponents because coupling affects local behavior only very slightly. Other coupling parameters are $ν = - 2$ and $θ = - 1.4$ (results almost identical if $ν = 0$ ). (b) Further evidence that Lyapunov exponents are not good indicators of synchronization. The exponents in this figure were obtained for two uncoupled chaotic neurons $(g_{c} = 0)$ . Changes in the exponents follow closely the changes in burst duty cycle (dashed) due to variations in excitation $σ$ . This happens because the system is expanding in the spiking intervals, and thus the average expanding rates are higher when spiking takes up a larger fraction of the total time, independently of synchronization.Reuse & Permissions
Figure 8
Phase plane of the chaotic Rulkov model coupled through (a) excitatory $(ν = 1)$ and (b) inhibitory $(ν = - 2)$ , fast threshold modulation synapses. Labels have the same meaning as in Fig. 3a, with primed versions for the shifted system of Eqs. (8) and unprimed for the isolated system of Eqs. (7). In both cases, $g_{c} = 0.3$ . Similar diagrams result from the nonchaotic Rulkov model.Reuse & Permissions
Figure 9
In-phase synchronization through excitatory $(ν = 1)$ FTM synapses in the nonchaotic Rulkov model. (a) When neuron 1 (solid) reaches $γ_{s n}$ , it begins to spike and drives neuron 2 (dashed) into spiking. Thus burst initiation is synchronized. (b) When neuron 2 (dashed) reaches $y = γ_{h}^{'}$ it jumps down into silence, driving neuron 1 (solid) into silence at the same time by shifting its nullclines. Thus burst ending is synchronized. Coupling parameters in this example are $g_{c} = 0.1$ and $θ = - 1.4$ . Excitation is $σ = - 0.9$ . A similar mechanism synchronizes bursts in the chaotic Rulkov model.Reuse & Permissions
Figure 10
Antiphase synchronization through inhibitory $(ν = - 2)$ FTM synapses in the nonchaotic Rulkov model. (a) When neuron 1 (solid) reaches $γ_{s n}$ , it begins to spike and switches neuron 2 (dashed) to its shifted nullclines. Thus burst initiation of neuron 1 delays burst initiation of neuron 2, favoring antiphase synchronization. (b) When neuron 1 (solid) reaches $y = γ_{h}$ it jumps down into silence; this switches neuron 2 (dashed) to its shifted nullclines and thus immediately releases it for bursting. Therefore the end of bursting in neuron 1 triggers the initiation of a burst in neuron 2. Coupling parameters in this example are $g_{c} = 0.1$ and $θ = - 1.4$ . Excitation is $σ = - 0.9$ . A similar mechanism synchronizes bursts in the chaotic Rulkov model.Reuse & Permissions
Figure 11
As a function of $g_{c}$ , and for three different values of synaptic reversal potential $ν$ : (a) Saddle-node $(Γ_{s n}^{'})$ and external crisis $(Γ_{c r}^{'})$ bifurcation curves of the shifted fast subsystem of Eqs. (8) in the chaotic Rulkov model. Vertical dotted lines indicate the position $γ_{s n}$ and $γ_{c r}$ of the isolated neuron bifurcations. (b) Saddle-node $(Γ_{s n}^{'})$ and homoclinic $(Γ_{h}^{'})$ bifurcation curves of the shifted fast subsystem of Eqs. (8) in the nonchaotic Rulkov model. Vertical dotted lines indicate the position $γ_{s n}$ and $γ_{h}$ of the isolated neuron bifurcations.Reuse & Permissions
Figure 12
Different synchronization regimes between two chaotic Rulkov neurons depending on the external excitation. In both cases, synapses have $ν = - 1.4$ , $g_{c} = 0.1$ , and $θ = - 1.4$ . In (a), $σ = - 1$ and antiphase synchronization sets in. In (b), $σ = - 1.5$ and in-phase synchronization dominates. The synchronization regime can be easily recognized by the parallelism or opposition of the slow variables. Apart from this, as it is to be expected, a low value of $σ$ , as in (b), produces sparse bursting while the high value of $σ$ in (a) gives rise to longer bursts.Reuse & Permissions
Figure 13
Effect of synapses on a chaotic Rulkov neuron depending on reversal potential: (a) $ν = - 1.4$ ; (b) $ν = - 1.6$ ; and (c) $ν = - 1.8$ . Two resting points $O_{1}$ and $O_{2}$ , corresponding to $σ_{1} = - 1.7$ (excitable neuron) and $σ_{2} = - 2$ (hyperpolarized neuron) are shown in each case. The arrows indicate the jump experienced by the neuron in the $η \to 0$ limit when the synapse is activated. In all cases, $g_{c} = 0.25$ . Labels for curves and bifurcation points are the same as in Fig. 8. Similar diagrams result from the nonchaotic model.Reuse & Permissions
Figure 14
Average cross-correlation of $x_{n, 1}$ and $x_{n, 2}$ of 50 trials of length $n_{max} = 50 000$ , as a function of excitation $σ$ for different values of synaptic threshold $ν$ . The dotted lines represent the 95% confidence interval of significance against the null hypothesis of uncoupled neurons. (a) Chaotic Rulkov model, with $g_{c} = 0.2$ and $θ = - 1.4$ . (b) Nonchaotic Rulkov model, with $g_{c} = 0.2$ and $θ = - 1.1$ . Note the sharp change in cross-correlation in the nonchaotic model at $σ = - 0.9$ when $ν = - 1.2$ .Reuse & Permissions
Figure 15
Contours of constant cross-correlation of $x_{n, 1}$ and $x_{n, 2}$ in the $σ − ν$ plane, for (a) the chaotic Rulkov model, and (b) for the nonchaotic Rulkov model. In both cases $g_{c} = 0.1$ . The contours were obtained from averages of 50 trials of length $n_{max} = 50 000$ in a grid of resolution $Δ ν = 0.04$ and $Δ σ = 0.02$ . The thick contour highlights zero cross-correlation.Reuse & Permissions
Figure 16
Contours of constant cross-correlation of $x_{n, 1}$ and $x_{n, 2}$ in (a) the $θ − ν$ plane, with $g_{c} = 0.1$ , and (b) the $g_{c} − ν$ plane, with $θ = 0$ , for the chaotic Rulkov model. The contours were obtained from averages of 50 trials of length $n_{max} = 50 000$ in a grid of resolution $Δ ν = 0.04$ , $Δ θ = 0.1$ , and $Δ g_{c} = 0.01$ . In both figures $σ = - 1.5$ . The thick contour highlights zero cross-correlation. Notice that the $g_{c}$ axis in (b) is inverted for easy comparison with the effects of $θ$ in (a).Reuse & Permissions
Figure 17
(a) Cross-correlation of $x_{n, 1}$ and $x_{n, 2}$ in the nonchaotic Rulkov model as a function of $ν$ and $θ$ , for $g_{c} = 0.25$ and $σ = - 1$ . Values of cross-correlation were obtained from averages of 50 trials of length $n_{max} = 50 000$ in a grid of resolution $Δ ν = 0.04$ and $Δ θ = 0.1$ . (b) Evolution of $x_{n, 1}$ (solid line) and $x_{n, 2}$ (dashed) at $ν = - 0.6$ , $g_{c} = 0.25$ , $σ = - 1$ , and $θ = 0.33$ : bursting is in-phase, spiking is antiphase inside each burst. (c) Same as in (b), but for $θ = 0.30$ : both bursts and spikes are synchronized in-phase, corresponding to a white spot of correlation 1 in (a). The orbit of neuron 2 has been shifted one sample in order to visualize it (otherwise, both orbits would be coincident).Reuse & Permissions
Figure 18
Boundary of the basin of attraction of synchronized spike orbits in the fast subsystem of the nonchaotic Rulkov model as a function of $γ_{1} = γ_{2} = γ$ . The basins are intervals starting at $x = - 1$ and ending at the depicted profiles. (a) Dependence of the basin on the synaptic threshold $θ$ . Other parameters are $ν = - 0.6$ and $g_{c} = 0.25$ . (b) Dependence of the basin on the synaptic reversal potential $ν$ . Other parameters are $θ = - 0.5$ and $g_{c} = 0.25$ . Note that the solid line in (b) is the same as the dotted line in (a).Reuse & Permissions
Figure 19
(a) Top: evolution of $x_{n, 1}$ (solid line) and $x_{n, 2}$ (dashed); bottom: evolution of $y_{n, 1}$ (solid line) and $y_{n, 2}$ (dashed), along a bursting orbit with $ν = - 0.6$ , $g_{c} = 0.25$ , $σ = - 1$ , and $θ = 0.5$ . In the first burst spikes have synchronized and the system is close to full synchronization, but during the second burst the slight mismatch between $y_{n, 1}$ and $y_{n, 2}$ is enough to destroy it. (b) Boundaries of the basin of attraction of synchronized spike orbits in the fast subsystem as a function of $γ_{1}$ for different values of $Δ γ = γ_{2} - γ_{1}$ and the same parameters as in (a). When $Δ γ > 0$ , the teeth of the profile are separated by gaps where spike synchronization is impossible.Reuse & Permissions
Figure 20
Raster plots of the bursting patterns in a ring of $N = 32$ nonchaotic Rulkov neurons. In all cases $α = 6$ , $η = 0.002$ , $g_{c} = 0.1$ , and $θ = - 1.1$ . External excitation $σ$ is $σ = - 1.2$ for $n = 1, \dots, 10 000$ and $σ = - 0.8$ for $n = 10 001, \dots, 20 000$ . (a) $ν = 0.0$ , in-phase burst synchronization. (b) $ν = - 1.2$ , switch between in-phase and antiphase synchronization. (c) $ν = - 2.0$ , antiphase synchronization.Reuse & Permissions
Figure 21
Raster plots of the bursting patterns in a ring of $N = 32$ chaotic Rulkov neurons. In all cases, $α = 4.15$ , $η = 0.001$ , $g_{c} = 0.1$ , and $θ = - 1.4$ . External excitation $σ$ is $σ = - 1.4$ for $n = 1, \dots, 10 000$ and $σ = - 1.0$ for $n = 10 001, \dots, 20 000$ . (a) $ν = 0.0$ , in-phase burst synchronization with occasional slips and propagation of activity. (b) $ν = - 1.45$ , switch between predominantly in-phase behavior (striped pattern) and predominantly antiphase behavior (checkered pattern). (c) $ν = - 2.0$ , antiphase propagation in the first half and antiphase synchronization in the second.Reuse & Permissions
Figure 22
Raster plot of the bursting patterns in a ring of $N = 32$ neurons with Hodgkin-Huxley dynamics connected by shunting $({GABA}_{A})$ synapses to their nearest neighbors. The model and parameters are as in [26], except that the ${GABA}_{A}$ synapses have a reversal potential of −60 mV. External excitation is $I = 0.05 μ A / {cm}^{2}$ during the first half of the simulation and $I = 0.35 μ A / {cm}^{2}$ during the second half. The network switches from a dominantly in-phase to a dominantly antiphase pattern.Reuse & Permissions
Figure 23
(a) Structure of a small regular network of 16 nonchaotic Rulkov neurons. Each neuron has six incoming synapses. Neurons have been divided in two groups (white and gray) according to the sign of their components in the dominant eigenvector of the adjacency matrix of the graph. (b) Activity in the network, corresponding to the last part of the raster in (c). We see neurons 1, 3, 4, 5, 7, and 11 firing in synchrony, and also 2, 6, 8, 10, 14, 15, and 16. They correspond to the two clusters shown in (a). Neurons 9, 12, and 13 are undecided between both clusters. (c) Raster plot of bursts in the network. In this simulation, $σ = - 1.2$ for $n = 1, \dots, 10 000$ and $σ = - 0.8$ for $n = 10 001, \dots, 20 000$ . In the first part, the whole network is synchronized approximately in-phase; in the second, neurons split in clusters that closely follow the division in two groups shown in (a), with frequent slips and readjustments. Other parameters are $α = 6$ , $η = 0.002$ , $g_{c} = 0.033$ , and $θ = - 1.05$ .Reuse & Permissions

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