Order parameter for bursting polyrhythms in multifunctional central pattern generators

We examine multistability of several coexisting bursting patterns in a central pattern generator network composed of three Hodgkin-Huxley type cells coupled reciprocally by inhibitory synapses. We establish that the control of switching between bursting polyrhythms and their bifurcations are determined by the temporal characteristics, such as the duty cycle, of networked interneurons and the coupling strength asymmetry. A computationally effective approach to the reduction of dynamics of the nine-dimensional network to two-dimensional Poincaré return mappings for phase lags between the interneurons is presented.

Figure 1

(a) Three-cell motif with asymmetric clockwise vs counterclockwise connection strengths. (b) Voltage traces: the phase (mod 1) of reference cell 1 (blue) is reset when $V_1$ reaches threshold ${\Theta }_{\mathrm{th}}=-40$ mV. The time delays between the burst onset in the reference cell 1 and the burst onset in cells 2 (green) and 3 (red), normalized over the recurrent time of cell 1, define a sequence of phase lags $\{\Delta {\varphi }_{21}^{(n)},\Delta {\varphi }_{31}^{(n)}\}$.

Figure 2

Five polyrhythms in the medium bursting motif at $g_{\mathrm{syn}}=5\times {10}^{-3}$ (increased to secure short transients for the purpose of illustration). Inhibitory pulses (horizontal bars) suppress the targeted cells, thus causing switching between co-existing rhythms: $(1\perp \{2\parallel 3\})$ in episode (i), traveling waves $(1\prec 2\prec 3)$ in (ii) and $(1\prec 3\prec 2)$ in (iii), followed by $(2\perp \{1\parallel 3\})$ led by cell 2 in (iv). Having released cells 1 and 2 simultaneously, cell 3 leads the motif in the $(3\perp \{1\parallel 2\})$ rhythm in the last episode.

Figure 3

(a) Phase-lag map for the symmetric, medium bursting motif showing five stable FPs: (red) dot at $~(\frac{1}{2},0)$, (green) $(0,\frac{1}{2})$, (blue) ($\frac{1}{2},\frac{1}{2}$), (black) $(\frac{2}{3},\frac{1}{3})$, and (purple) ($\frac{1}{3},\frac{2}{3}$), corresponding to the antiphase $(3\perp \{1\parallel 2\})$, $(2\perp \{1\parallel 3\})$, $(1\perp \{2\parallel 3\})$ bursts, and traveling clockwise $(1\prec 2\prec 3)$ and counterclockwise $(1\prec 3\prec 2)$ waves; the attraction basins are divided by “separatrices” (stable sets) of six saddles (small grey dots). Arrows indicate the forward iterate direction in the phase plane. (b) Asymmetric motif at $g_{\circlearrowright }=0.154$ near the saddle-node bifurcations annihilating three stable FPs for antiphase bursting rhythms.

Figure 4

(a) Phase-lag map for the symmetric, short bursting motif showing only three stable FPs [blue dot at ($\frac{1}{2},\frac{1}{2}$), red dot at $(\frac{1}{2},0)$, and green dot at $(0,\frac{1}{2})$] corresponding to antiphase rhythms where one cell bursts followed by synchronized bursts in the other two cells. Unstable FPs at $(\frac{2}{3},\frac{1}{3})$ and ($\frac{1}{3},\frac{2}{3}$) exclude the clockwise $(1\prec 2\prec 3)$ and counterclockwise $(1\prec 3\prec 2)$ traveling waves from the repertoire of the short bursting motif. (b) Map for the asymmetric motif ($g_{\circlearrowright }=0.2$) depicting a stable invariant curve near a three-saddle connection around the FP at ($\frac{2}{3},\frac{1}{3}$).

Figure 5

(a) Phase-lag map for the symmetric, long bursting motif revealing two equally dominant attractors: ($\frac{2}{3},\frac{1}{3}$) and ($\frac{1}{3},\frac{2}{3}$) for $(1\prec 2\prec 3)$ and $(1\prec 3\prec 2)$ traveling rhythms. (b) Map for the asymmetric ($g_{\circlearrowright }=0.3$) medium bursting motif depicting two persistent attractors: the one for the clockwise $(1\prec 2\prec 3)$ rhythms prevails over the attractor for the counterclockwise $(1\prec 3\prec 2)$ rhythm. Further increase in $g_{\circlearrowright }$ leads to the only observable $(1\prec 2\prec 3)$ rhythm, after FP at ($\frac{1}{3},\frac{2}{3}$) merges with three nearby saddles.