Effect of Electrical Synapses in the Cycle-by-Cycle Period and Burst Duration of Central Pattern Generators

Abstract

Central Pattern Generators (CPGs) are neural circuits that generate robust coordinated neural activity to control motor rhythms. Many CPGs are convenient neural circuits for locomotion control in autonomous robots. In this context, invertebrate CPGs are key networks to understand rhythm generation and coordination, as their cells and connections can be identified and mapped, like in the crustacean pyloric CPG. Experiments during the last decades have shown that mutual inhibition by chemical synapses together with electrical coupling underlie the timing of neuron activations that shape each rhythm cycle of this CPG. Due to the presence of inhibitory and electrical synapses, regular and irregular triphasic spiking-bursting activity can be found in the pyloric CPG, always preserving the same neuron activation sequence. In this study, we use a model of this well-known CPG to assess the role of electrical synapses in shaping the cycle-by-cycle period and individual cell burst duration. We show that electrical coupling strength asymmetrically affects the burst duration of each individual neuron, as well as the overall cycle-by-cycle duration. Our results support the view that electrical coupling largely contributes to shape the intervals that define functional sequences in CPGs, which can be applied in bioinspired autonomous robotic motor control.

Appendices

A Equations of Komendantov-Kononenko (KK) Model

The equation that describes the membrane potential in this model is:

$$\begin{aligned} - C_mdV(t)/dt = I_{Na(TTX)}+I_{K(TEA)}+I_{K}+I_{Na} +I_{Na}(V)+I_{B}+I_{Ca}+I_{Ca-Ca} +I_{Syn} \end{aligned}$$

(5)

where $C_m$ is the capacitance of the membrane and V the membrane potential in mV. $I_{Syn}$ represents the total synaptic current, electrical or chemical, which is different for each neuron in the CPG.

The model is defined with four components describing distinct ionic currents:

• 1. A slow wave generating mechanism, given by sodium, potassium and chemosensitive currents:

  $$\begin{aligned} I_{Na}(V)=g^*_{Na}(V)(1/1+\exp {(-0.2(V(t)+45))})) (V(t)-V_{Na})); \end{aligned}$$

  (6)
  $$\begin{aligned} I_{Na}=g^*_{Na}(V(t)-V_{Na}); \end{aligned}$$

  (7)
  $$\begin{aligned} I_{K}=g^*_{K}(V(t)-V_{K}); \end{aligned}$$

  (8)
  $$\begin{aligned} I_{B}=g^*_{B}m_B(t)h_B(t)(V(t)-V_{B}); \end{aligned}$$

  (9)
  $$\begin{aligned} dm_B(t)/dt=(1/(1+\exp {(0.4(V(t)+34))})-m_B(t))/0.05; \end{aligned}$$

  (10)
  $$\begin{aligned} dh_B(t)/dt=(1/(1+\exp {(-0.55(V(t)+43))})-h_B(t))/1.5; \end{aligned}$$

  (11)

  where $m_B(t)$ and $h_B(t)$ are conductance variables that describe the activation and deactivation of the ionic conductance. The parameters $g^*_{Na}(V)$, $g^*_{Na}$, $g^*_{K}$, $g^*_{B}$ are the maximal conductances of these ionic channels and $V_{Na}$, $V_{K}$ and $V_{B}$ are the corresponding reversal potentials. We will use the same notation for the channels described bellow.

• 2. A spike-generating mechanism, which is described by TTX-sensitive sodium and TEA-sensitive potassium Hodgkin-Huxley type currents:

  $$\begin{aligned} I_{Na(TTX)}=g^*_{Na(TTX)}m^3(t)h(t)(V(t)-V_{Na}); \end{aligned}$$

  (12)
  $$\begin{aligned} I_{K(TEA)}=g^*_{K(TEA)}n^4(t)(V(t)-V_{K}) \end{aligned}$$

  (13)
  $$\begin{aligned} dm(t)/dt=(1/(1+\exp {(-0.4(V(t)+31))})-m(t))/0.0005; \end{aligned}$$

  (14)
  $$\begin{aligned} dh(t)/dt=(1/(1+\exp {(0.25(V(t)+45))})-h(t))/0.01; \end{aligned}$$

  (15)
  $$\begin{aligned} dn(t)/dt=(1/(1+\exp {(-0.18(V(t)+25))})-n(t))/0.015; \end{aligned}$$

  (16)

• 3. A calcium transient voltage-dependent current, described by:

  $$\begin{aligned} I_{Ca}=g^*_{Ca}m^2_{Ca}(V(t)-V_{Ca}); \end{aligned}$$

  (17)
  $$\begin{aligned} dm_{Ca}(t)/dt=(1/(1+\exp {(-0.2(V(t))})-m_{Ca}(t))/0.01; \end{aligned}$$

  (18)

• 4. A calcium stationary $[Ca^{2+}]_{in}$ inhibited current given by:

  $$\begin{aligned} I_{Ca-Ca} =g^*{Ca-Ca}\frac{1}{1+\exp {(-0.06(V(t)+45))}} \frac{1}{1+\exp {(K_{\beta }([Ca](t)-\beta ))}}(V(t)-V_{Ca}); \end{aligned}$$

  (19)
  $$\begin{aligned} d[Ca](t)/dt=\rho -I_{Ca}/2F\nu -K_s[Ca](t); \end{aligned}$$

  (20)

  where $\nu = 4\pi R^3/3$ is the volume of the cell; [Ca] is $[Ca^{2+}]_{in} \, (mM)$, F is Faraday number ($96,485 \,$ Cmol$^{-1}$), $K_s$ is the rate constant of intracellular Ca-uptake by intracellular stores and $\rho$ is the endogenous Ca buffer capacity.

B Parameters Used in the KK Model

• $V_{Na} = 40$ mV; $V_{K} = -70$ mV; $V_{B} = -58$ mV; $V_{Ca} = 150$ mV; $C_{m} = 0.02\, \upmu$F; $R = 0.1$ mm; $k_{s} = 50$ l/s; $\rho = 0.002$; $k_{\beta } = 15000$ mV; $\beta = 0.00004$ mM;

• Chaotic mode: $g^*_{K} = 0.25$  $\upmu$S; $g^*_{Na} = 0.0231$  $\upmu$S; $g^*_{Na}(V) = 0.11$  $\upmu$S; $g^*_{B} = 0.1372$ $\upmu$S; $g^*_{Na(TTX)} = 400$ $\upmu$S; $g^*_{K(TEA)} = 10$ $\upmu$S; $g^*_{Ca} = 1.5$ $\upmu$S; $g^*_{Ca-Ca} = 0.02$ $\upmu$S;

• Chaotic bursting mode: $g^*_{K} = 0.25$ $\upmu$S; $g^*_{Na} = 0.02$ $\upmu$S; $g^*_{Na}(V) = 0.13$ $\upmu$S; $g^*_{B} = 0.18$ $\upmu$S; $g^*_{Na(TTX)} = 400$ $\upmu$S; $g^*_{K(TEA)} = 10$ $\upmu$S; $g^*_{Ca} = 1$ $\upmu$S; $g^*_{Ca-Ca} = 0.01$ $\upmu$S;

• Regular mode: $g^*_{K} = 0.25$ $\upmu$S; $g^*_{Na} = 0.02$ $\upmu$S; $g^*_{Na}(V) = 0.13$ $\upmu$S; $g^*_{B} = 0.165$ $\upmu$S; $g^*_{Na(TTX)} = 400$ $\upmu$S; $g^*_{K(TEA)} = 10$ $\upmu$S; $g^*_{Ca} = 1$ $\upmu$S; $g^*_{Ca-Ca} = 0.01$ $\upmu$S;

• Initial conditions: $V = -55$ mV; $[Ca^{2+}]_{in}= 0$;