Generalized half-center oscillators with short-term synaptic plasticity

How can we develop simple yet realistic models of the small neural circuits known as central pattern generators (CPGs), which contribute to generate complex multiphase locomotion in living animals? In this paper we introduce a new model (with design criteria) of a generalized half-center oscillator, (pools of) neurons reciprocally coupled by fast/slow inhibitory and excitatory synapses, to produce either alternating bursting or other rhythmic patterns, characterized by different phase lags, depending on the sensory or other external input. We also show how to calibrate its parameters, based on both physiological and functional criteria and on bifurcation analysis. This model accounts for short-term neuromodulation in a biophysically plausible way and is a building block to develop more realistic and functionally accurate CPG models. Examples and counterexamples are used to point out the generality and effectiveness of our design approach.

Figure 1

gHCO neural circuit with inhibitory (denoted with $▸◂$) and excitatory ($•$) synapses reciprocally coupling two oscillatory cells.

Figure 2

Asymptotic antiphase (a) and synchronous (b) bursting voltage traces $V_1$ (red) and $V_2$ (blue) at $I_c=-0.43$ and 0.13, respectively, in gHCO (1–3), superimposed with excitatory/inhibitory thresholds ${\theta }^{}$ (horizontal lines) at 25 and $-30$ mV. (c, d) Synapse dynamics: fast modulatory $s_2^{\text{in}}\left(t\right)$ (gray) vs. slowly summating/decaying $s_2^{\text{ex}}\left(t\right)$ (black). See the Appendix for parameters.

Figure 3

Bifurcation diagram showing how the phase-lag $\mathrm{\Delta }$ between the gHCO neurons is affected by the current $I_c$; here, 30 initial $\mathrm{\Delta }$-values were sampled evenly between 0.05 and 0.95 for each of the 50 $I_c$-values. Parameters listed in the Appendix.

Figure 4

(a) Asymptotic bursting voltage trace with undershoot produced by the Plant neuron model [39, 40]. (b) Voltage traces produced by the gHCO with two coupled Plant neurons. See the Appendix for parameters.

Figure 5

(a) Mean values (over 5 s) of the IBI (green line) and the burst duration (black line) plotted against $I_{\text{ext}}$ for the exponential IF-model [41]. Corresponding bifurcation diagram for the phase lag $\mathrm{\Delta }$ between the cells in the gHCO, in which each cell is an exponential IF-model (b).

Figure 6

(a) Mean values (over 5 s) of the IBI plotted against $g_e$ for the exponential IF-model [41]. Corresponding bifurcation diagram for the phase lag $\mathrm{\Delta }$ between the cells in the gHCO, in which each cell is an exponential IF-model (b).

Figure 7

Bifurcation diagram showing the flat-even phase-lags, $\mathrm{\Delta }=\{0,1\}$ (in-phase) and $\mathrm{\Delta }=0.5$ (antiphase), between the bursters plotted against the current $I_c$ for the gHCO with slow inhibitory and fast excitatory synapses; here, 30 initial $\mathrm{\Delta }$-values were sampled evenly between 0.05 and 0.95 for every $I_c$ value out of 50. Parameters listed in the Appendix.

Figure 8

(a) Time evolution of the phase lag $\mathrm{\Delta }$ between the gHCO cells (black line) in response to step-wise changes of $I_c$ (green dashed lines); $I_c$ increased over 25 steps from $-0.43$ to 0.13, only the time window in which $\mathrm{\Delta }$ transition occurs is shown. (b) Voltage traces progressing from in-phase to antiphase bursting within the time window bounded by the grey vertical lines in panel (a).