Experimental observation of multistability and dynamic attractors in silicon central pattern generators

We report on the multistability of chaotic networks of silicon neurons and demonstrate how spatiotemporal sequences of voltage oscillations are selected with timed current stimuli. A three neuron central pattern generator was built by interconnecting Hodgkin-Huxley neurons with mutually inhibitory links mimicking gap junctions. By systematically varying the timing of current stimuli applied to individual neurons, we generate the phase lag maps of neuronal oscillators and study their dependence on the network connectivity. We identify up to six attractors consisting of triphasic sequences of unevenly spaced pulses propagating clockwise and anticlockwise. While confirming theoretical predictions, our experiments reveal more complex oscillatory patterns shaped by the ratio of the pulse width to the oscillation period. Our work contributes to validating the command neuron hypothesis.

Figure 1

(a) Electrical circuit of the silicon neuron. $C$ is the membrane capacitance; $g_{\text{Na}},g_{\text{K}}$, and $g_{\text{L}}$ are the Na, K, and leakage conductances; $E_{\text{Na}}$ and $E_{\text{K}}$ are the reversal potentials. The injection of a depolarizing current $I_{\text{stim}}$ induces oscillations of the membrane voltage $V$. (b) Membrane voltage oscillations evoked by current stimulation of amplitude $60 \mu \mathrm{A}$ (black curves, labeled 1), $80 \mu \mathrm{A}$ (red curves, 2), and $100 \mu \mathrm{A}$ (blue curves, 3). Main panel: Excitatory response with current threshold $I_{\text{th}}=86\mu \mathrm{A}$.

Figure 2

(a) Inhibitory link injecting a hyperpolarizing current $I_{\text{post}}=g\left(V_{\text{max}}\right)(V_{\text{post}}-V_{\text{pre}})$ in the postsynaptic neuron. (b) Experimental dependence of the postsynaptic current on $V_{\text{post}}-V_{\text{pre}}$ at different values of $V_{\text{max}}$. (c) Dependence of the link conductance on $V_{\text{max}}$. (d) Oscillations of a neuron pair coupled via reciprocally excitatory (top) and reciprocally inhibitory links (bottom). $I_{\text{stim}}=100\mu \mathrm{A},g_{12}=23.5\mu \mathrm{S}$, and $g_{21}=37.2\mu \mathrm{S}$.

Figure 3

(a) Timed current steps applied to neurons 1, 2, and 3. Neurons also receive postsynaptic currents from the reciprocally inhibitory links interconnecting them. (b) Circuit board.

Figure 4

(a) Transient voltage oscillations of individual neurons stimulated by $100 \mu \mathrm{A}$ current steps delayed by $\mathrm{\Delta }{t}_{12}=0.6T$ and $\mathrm{\Delta }{t}_{13}=0.7T$. The index $n$ counts the interspike intervals of neuron 1. (b) Phase response trajectory plotting the dephasing between neurons 3 and 1 as a function of the dephasing between neurons 2 and 1. Initial conditions are the same as in (a). (c) Trajectory of the dynamic state evolving towards a limit cycle (red closed loop). Synaptic conductance: $g_{12}=g_{23}=g_{31}=16\mu \mathrm{S}$ and $g_{21}=g_{13}=g_{32}=45\mu \mathrm{S}$.

Figure 5

(a) Phase lag map of a three neuron CPG with reciprocally inhibitory links projecting predominantly in the anticlockwise direction. The initial conditions for each trajectory are set by the relative timings $\mathrm{\Delta }{t}_{12}$ and $\mathrm{\Delta }{t}_{13}$ which we vary from 0 to $T$ in steps of $0.05T$. $T=2.80$ ms is the period of the oscillations of neuron 1 in the steady state. Two attractors are observed at $3⥁ (0.16,0.49)$ and $2⥀ (0.62,0.30)$ which correspond to clockwise and anticlockwise firing sequences. (b) Wave forms of attractors $3⥁$ and $2⥀$. Synaptic conductances: $g_{12}=g_{23}=g_{31}=16 \mu \mathrm{S}$ and $g_{21}=g_{13}=g_{31}=45 \mu \mathrm{S}$.

Figure 6

(a) Phase lag map obtained by weakening the asymmetry of reciprocally inhibitory links through increases in clockwise inhibition from $16\mu \mathrm{S}$ to $g_{12}=23\mu \mathrm{S},g_{23}=g_{31}=20\mu \mathrm{S}$, and decreasing anticlockwise inhibition from $45\mu \mathrm{S}$ to $g_{21}=37\mu \mathrm{S},g_{13}=45\mu \mathrm{S},g_{32}=39\mu \mathrm{S}$. The period of neuron 1 remains $T=2.80$ ms. Three attractors are observed at the coordinates $3⥁ (0.21,0.56),1⥁ (0.43,0.61)$, and $2⥀ (0.74,0.34)$. (b) Spatiotemporal oscillations corresponding to the three attractors.

Figure 7

Dependence of phase lag maps on the amplitude of current stimulation. The network connectivity is identical to Fig. 6 but the current amplitude is increased from $I_{\text{stim}}=100$ to $120\mu \text{A}$. Oscillation period: $T=2.2$ ms.

Figure 8

(a) Rhythms in which two neurons fire in rapid succession out of phase with the third. The black, red, and blue numbered blocks show the positions of spikes 1, 2, and 3 within one period of the cycle. $\zeta =W/T$ is the spike width normalized by the oscillation period. (b) Phase lag map showing the position of the theoretically predicted attractors (dots in the background) with the experimentally observed ones (dots in the foreground).

Figure 9

(a) Anticlockwise and (b) clockwise triphasic bursting patterns produced by three Hindmarsh-Rose neurons coupled via mutually inhibitory links, $I_{\text{post}}=g(V_{\text{post}}-V_{\text{pre}})$, where $g>0$.