Robust design of polyrhythmic neural circuits

Neural circuit motifs producing coexistent rhythmic patterns are treated as building blocks of multifunctional neuronal networks. We study the robustness of such a motif of inhibitory model neurons to reliably sustain bursting polyrhythms under random perturbations. Without noise, the exponential stability of each of the coexisting rhythms increases with strengthened synaptic coupling, thus indicating an increased robustness. Conversely, after adding noise we find that noise-induced rhythm switching intensifies if the coupling strength is increased beyond a critical value, indicating a decreased robustness. We analyze this stochastic arrhythmia and develop a generic description of its dynamic mechanism. Based on our mechanistic insight, we show how physiological parameters of neuronal dynamics and network coupling can be balanced to enhance rhythm robustness against noise. Our findings are applicable to a broad class of relaxation-oscillator networks, including Fitzhugh-Nagumo and other Hodgkin-Huxley-type networks.

Figure 1

Bursting in the slow-fast Hodgkin-Huxley neuronal model. The bursting orbit of a single neuronal burster (at $\sigma =0$ and $g_{\mathrm{inh}}=0$) is organized according to the backbone of nullclines for the slow variable, given by ${\dot{m}}_i=0$ [dashed (red) line], and fast variables $({\dot{V}}_i,{\dot{h}}_i)=0$ (dashed-dotted black lines). The shaded rectangle (lower-left corner) is expanded in Fig. 3.

Figure 2

Stable polyrhythmic patterns in the NCM. Two $0.5$-mV kicks (arrows) applied to membrane voltages $V_i\left(t\right)$ cause the NCM to switch among the three coexistent pacemaker patterns (A, B, and C). Parameters are $\sigma =0$, $g=20$ pS.

Figure 3

Critical synaptic strength in the neuronal burster. We show the shaded region of state space from Fig. 1 (lower-left corner there). Constant inhibition, at coupling strengths $g_{\mathrm{inh}}>g_{\mathrm{crit}}^{*}$, induces a saddle-node bifurcation by shifting the fast nullcline (dashed-dotted black lines) across the slow one [dashed (red) line]. The critical value $g_{\mathrm{crit}}^{*}$, at which nullclines are tangent (filled circle), therefore separates a soft coupling from a hard coupling that can lock down the postsynaptic burster.

Figure 4

(a) NCM of three bursters (blue, 1; green, 2; red, 3) randomly switches among three pacemaker patterns in the voltage trace for coupling strength $g_{\mathrm{inh}}=15$ pS and noise ${\sigma }^2=0.0025{\text{pA}}^2$/s. (b) Coincident bursts (shaded regions) are mapped into shifts in the A-B-C directions of 2D random walks. Inset: Random walk episode corresponding to the voltage trace in (a). The mean free path of the trajectory is $3.8$ steps.

Figure 5

Nonmonotonous dependence of the mean free path (MFP) on the synaptic strength $g_{\mathrm{inh}}$. For a plausible range of noise intensities, ${\sigma }^2$, the MFP reveals a synaptic strength of maximal robustness, $g_{\mathrm{opt}}\simeq 5.5$ pS, comparable with the critical coupling $g_{\mathrm{crit}}=6.1$ pS [Eq. (10)].

Figure 6

Hard-lock mechanism of rhythm switching. (a) A typical rhythm switching event (inlet) occurs upon noise-induced separation of neurons 1 and 2. Neuron 1 reaches $\Theta$ and inhibits neuron 2 from bursting. (b) The quiescent phase of postsynaptic neurons undergoes a saddle-node bifurcation upon activation of inhibition (left panel). The location of the unstable point marks the critical voltage $V_{\mathrm{crit}}$ [Eq. (11)] separating rhythm switching from coincident bursting: in the right panel, model neuron 2 [Eq. (9)] stays below $V_{\mathrm{crit}}$, thus switching rhythms. Parameters: $g_{\mathrm{inh}}=20$ pS, ${\sigma }^2=0.01{\text{pA}}^2/\text{s}$, $V_0=-44.3\text{mV}$, $\varepsilon =0.22\text{mV}/\text{s}$, $\alpha =1.53{\text{mV}}^{-1}{\text{s}}^{-1}$.

Figure 7

Estimation procedure of the saddle-node ghost equation. (a) Time derivative $\dot{V}$ on the periodic orbit shows a complicated dependence on $V$. (b) Locally, $\dot{V}$ can be expressed as a function, $F\left(V\right)=\dot{V}$. Parameters $\varepsilon$, $V_0$, and $\alpha$ of Eq. (9) are determined so that the quadratic fit (dash-dotted line) matches $F\left(V\right)$ (solid line) at the local minimum of the quiescent period. Ghost model parameters: $V_0=-44.3\text{mV}$, $\epsilon =0.22\text{mV}/\text{s}$, $\alpha =1.53{\text{mV}}^{-1}{\text{s}}^{-1}$.

Figure 8

Improving the robustness of bursting polyrhythms. (a) MFP dependence on $g_{\mathrm{inh}}$ for $\varepsilon =0.22$ mV/s at $V^{{\mathrm{K}}_2}=3$ mV, $0.25$ mV/s at $3.5$ mV, and $0.27$ mV/s at 4 mV. The optimum, $g_{\mathrm{opt}}$, shifts towards a higher $g_{\mathrm{inh}}$. (b) MFP dependence on $g_{\mathrm{inh}}$ in the NCM with instantaneous $\tau =0$ and delayed synapses $\tau =100$ and 250 ms. Parameters: $V^{{\mathrm{K}}_2}=3$ mV, ${\sigma }^2=0.0025{\text{pA}}^2/\mathrm{s}$.

Figure 9

Comparison of critical and optimal inhibitory strength. (a) For different values of $V^{{\mathrm{K}}_2}$, $g_{\mathrm{crit}}$ [dashed-dotted (blue) line] and $g_{\mathrm{crit}}^{*}$ [dashed (red) line] approximate the optimal coupling $g_{\mathrm{opt}}$ (circles) reasonably well. (b) The ghost model approximation $g_{\mathrm{crit}}$ systematically underestimates the bifurcation value $g_{\mathrm{crit}}^{*}$, seen as deviations from the diagonal (dashed black line). Parameter: ${\sigma }^2=0.0025{\text{pA}}^2/\mathrm{s}$.