When two wrongs make a right: synchronized neuronal bursting from combined electrical and inhibitory coupling

Abstract

Synchronized cortical activities in the central nervous systems of mammals are crucial for sensory perception, coordination and locomotory function. The neuronal mechanisms that generate synchronous synaptic inputs in the neocortex are far from being fully understood. In this paper, we study the emergence of synchronization in networks of bursting neurons as a highly non-trivial, combined effect of electrical and inhibitory connections. We report a counterintuitive find that combined electrical and inhibitory coupling can synergistically induce robust synchronization in a range of parameters where electrical coupling alone promotes anti-phase spiking and inhibition induces anti-phase bursting. We reveal the underlying mechanism, which uses a balance between hidden properties of electrical and inhibitory coupling to act together to synchronize neuronal bursting. We show that this balance is controlled by the duty cycle of the self-coupled system which governs the synchronized bursting rhythm.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

Keywords: bursting neurons; duty cycle; electrical coupling; inhibition; synchronization.

Figure 1.

Square-wave bursting in the uncoupled Sherman model (2.1) with g_inh=0 (nS) and g_el=0 (nS). (Main graph) The dotted curve schematically indicates the route for the bursting solution. The plane V =Θ_s= −40 (mV) corresponds to the synaptic threshold. (Inset) Corresponding time series of square-wave bursting. (Online version in colour.)

Figure 2.

The combined effect of electrical and inhibitory synapses on complete synchronization in the two-cell network. The colour bar indicates the voltage difference ΔV =|V₁−V₂| (mV), averaged over the last three bursting periods. The black (blue) zone corresponds to the zero voltage difference (complete synchronization). The dark grey (red) colour indicates anti-phase bursting with the maximum voltage difference (approx. 40 mV). (a,b) Established phase locking from initial conditions where the first cell is in the active spiking phase while the second is silent (a) and initial conditions close to complete synchrony (b). (c,d). Zoom-ins of the corresponding top diagrams. Coexistence of synchronized and anti-phase bursting. The scattered black (blue) regions (c) correspond to the onset of complete synchronization from the unfavourable initial conditions. The synchronization effect is much more pronounced when the cells start from close initial conditions, as indicated by the black (blue) tongue-shaped region (d). Parameters corresponding to points A (g_inh=0.0001;g_el=0.0001 (nS)); B (g_inh=0;g_el=0.01 (nS)); C (g_inh=0.01;g_el=0.01 (nS)); D (g_inh= 0.01; g_el=0 (nS)); E (g_inh=0.02; g_el=0.01 (nS)); and F (g_inh= 0.01; g_el=0.02 (nS)). (Online version in colour.)

Figure 3.

Largest transversal Lyapunov exponent λ_⊥ for the stability of the synchronous solution in the two-cell network with purely electrical connections (g_in=0). Positive (negative) values indicate instability (stability) of synchronization. Increasing g_el from 0 first makes the electrical coupling desynchronize the cells within a range of moderate coupling (see the zoomed region g_el∈(0,0.02), where the dependence of λ_⊥ on g_el is monotonic). Any further increase in g_el beyond 0.04 makes the electrical coupling synchronizing, as the Lyapunov exponent becomes less positive. The zoomed region corresponds to the heatmap in figure 2c,d. (Online version in colour.)

Figure 4.

Poincaré maps for the evolution of the phase difference in the two-cell network and the corresponding voltage traces. Initial phase differences Δϕ_n (horizontal axes) versus the phase differences after k bursts Δϕ_n+k, with k=40. The phase difference is normalized to 1, where the zero phase difference Δϕ=0 corresponds to complete synchrony and Δϕ=0.5 indicates anti-phase bursting. Intersections of the graph Φ(Δϕ) (solid curve) with the diagonal (dashed) line yield phase-locked states. (a) Graphs A, B, C and E correspond to points A, B, C and E in figure 2. (A) Weak electrical and inhibitory synapses yield multiple phase-locked states as fixed points of the phase map. These include stable anti-phase bursting (star), complete synchrony (solid circle) and an unstable state at Δϕ_n≈0.015 which separates the attraction basins of the stable states. Note a much larger attraction basin of anti-phase bursting. Arrowed lines on the cobweb diagram illustrate the convergence to the anti-phase state from a given initial condition (a). Voltage traces of coexisting anti-phase bursting and complete synchrony (b). (B) Electrical coupling induces phase-locking with a small phase difference between the bursts; however, the spikes within the bursts are in anti-phase. (C) Stable complete synchronization with a large basin of attraction. (E) Phase-locking with Δϕ≈0.4, close to anti-phase bursting. The cloud of dots rather than a baseline phase-shift curve originates from varying duty cycles of the cells and numerical difficulties in identifying the initial ratio of the burst period over the phase shift to the terminal ratio of the same quantities. (Online version in colour.)

Figure 5.

Transformation of the full coupled system into two subsystems: fast (a) and slow (b). Fixing the slow variable S at a given value, S=0.18, turns the coupled system into a network of interacting tonic spiking cells. This fast system accounts for the interaction in the full system when both cells are in the spiking phase. Ignoring the spikes transforms the coupled system into a network of two slow relaxation oscillators, which mimics the interaction between the cells at the level of bursts (envelopes of spikes). (Online version in colour.)

Figure 6.

The effect of electrical and inhibitory synapses on the synchronization properties of the dissected, fast (a) and slow (b) subsystems. Electrical and inhibitory synapses play opposite roles in promoting synchrony in the fast and slow subsystems. When isolated, electrical synapses promote anti-phase spiking in the coupled fast system (a,c) and synchrony in the slow system (b,d). Inhibitory coupling induces spike synchrony in the fast subsystem and fosters anti-phase bursting in the slow one. (a,b) Heatmaps and colour-coding are similar to those of figure 2. The circle and the triangle correspond to point B in figure 2. The square and diamond indicate point D in figure 2. (c,d) The corresponding voltage traces. (Online version in colour.)

Figure 7.

Stabilizing and destabilizing components of the inhibitory coupling. (a) Voltage trace of 12-spike synchronous bursting. The horizontal line indicates the synaptic threshold Θ_s=−40 (mV), above which the inhibition activates. (b) The synaptic term S₁≥0, which promotes spike synchrony, is turned on during the duration of a spike. (c) The destabilizing synaptic term S₂≤0 is on during instances when the voltage crosses the synaptic threshold. (Online version in colour.)

Figure 8.

Largest transversal Lyapunov exponent λ_⊥, the duty cycle of synchronous bursting, and the averaged synaptic terms 〈S₁+S₂〉 as a function of the inhibitory coupling (electrical coupling g_el=0.01 is fixed; this diagram corresponds to the B−C−E route in figure 2). The dotted line (a) is zero and hence represents the transition to stable synchrony, which occurs when λ_⊥<0. The sign of λ_⊥ changes at the values g_in≈0.009 and g_in≈0.017, which bound the stable region (see figure 2) and are indicated by the vertical lines in each plot. The duty cycle of the self-coupled system (2.2), which governs the synchronous solution, reaches its minimal values within the stable region (b(i)). The shorter duty cycle yields maximal values of the synaptic terms S₁+S₂ averaged over one period of oscillations (b(ii)), such that the overall stabilizing effect of the inhibitory coupling can stabilize the synchronous solution. The sharp drop (rise) in the size of the duty cycle (synaptic terms) is due to the addition of one spike in the burst. (Online version in colour.)

Figure 9.

Stability diagrams for network synchronization in four-cell networks, similar to the heatmaps of figure 2. The corresponding topologies illustrated underneath each figure; spring-like (solid circle) lines indicate electrical (inhibitory) connections. The colour bar depicts the mean voltage difference (mV), calculated and averaged over four bursting periods. The black (blue) bounded region represents complete synchrony. Note the maximal area of stable synchrony in the network with both local electrical and inhibitory connections (d); this indicates that the combined synergistic effect is strongest in sparse configurations with connected graphs. (Online version in colour.)

Figure 10.

(a) Thirty-cell random network with electrical and inhibitory connections. The structure of directed inhibitory connections (thin (blue) lines) is random, with a constraint on the uniform node degree k_inh=8. Undirected electrical connections (thick (red) lines) are randomly generated; the node degree ranges randomly from 1 to 15. (b) The stability diagram and colour coding are similar to those of figure 9. Note the presence of the combined effect of electrical and inhibitory coupling. (Online version in colour.)