Temperature-modulated synchronization transition in coupled neuronal oscillators

We study two firing properties to characterize the activities of a neuron: frequency-current ($f$-$I$) curves and phase response curves (PRCs), with variation in the intrinsic temperature scaling parameter ($\mu$) controlling the opening and closing of ionic channels. We show a peak of the firing frequency for small $\mu$ in a class I neuron with the $I$ value immediately after the saddle-node bifurcation, which is entirely different from previous experimental reports as well as model studies. The PRC takes a type II form on a logarithmic $f$-$I$ curve when $\mu$ is small. Then, we analyze the synchronization phenomena in a two-neuron network using the phase-reduction method. We find common $\mu$-dependent transition and bifurcation of synchronizations, regardless of the values of $I$. Such results give us helpful insight into synchronizations tuned with a sinusoidal-wave temperature modulation on neurons.

Figure 1

Dynamics of the Morris-Lecar-type spiking model and synaptic potential for $V$, $W$, and $s$. Here we used the parameter set of $\mu =0.25$, $I=39.695$ $\mu$A/cm${}^2$, and ${\tau }_{\mathrm{syn}}=2.5$ ms.

Figure 2

Trajectories on $V$-$W$ plane diagrams and time evolutions of membrane potential for the ML model with $I=39.7$ influenced with a temperature scaling factor $\mu$. (a) The whole nullclines of $dV/dt=0$ (a solid line) and $dV/dt=0$ (a broken line) depicted on the $V$-$W$ plane. In (b), the temporal duration between points indicates a velocity of $V$. The $V$ motion on the active or inactive phases gradually becomes slower. $\Phi$ is a variable normalized by periodicity. (c) The $V$-$W$ plane (a) is scaled up toward the saddle-node bifurcation point. (d) Plots of $\mu$ on $dV/dt$ and $-I_{Ca}$.

Figure 3

Simulation results of the hyperpolarization process in the ML model. (a) Trajectories, which are controlled with three different temperature scaling factors $\mu =2.0$, $1.0$, and $0.1$, are drawn when they start near an unstable fixed point on the $V$-$W$ phase plane. (b) Such hyperpolarization demonstrations also shown with time evolutions of the membrane potential. Here $I=39.7$.

Figure 4

Temperature-scaling-factor-dependent firing frequency $f$ vs current ($I$) relationship for the ML model. (a) An $I$-$\mu$ diagram for $f$. (b) Different $f$-$I$ curves with $\mu =1.0$, $0.25,$ and $0.20$. Broken lines show functions of $f$ as $I-I_b$, where $I_b$ is near the bifurcation. (c) $f$-$\mu$ curves at fixed $I$.

Figure 5

($\mu$,$I$)-dependent PRCs for the class I type ML model, corresponding to the membrane potential dynamics. (a) represents a color map for a firing frequency, in terms of $I$ and $\mu$. (b) to (e) show PRCs and membrane potential dynamics. (b) $I=39.6732$ and $\mu =0.2$. (c) $I=45.0$ and $\mu =0.2$. (d) $I=39.694$ and $\mu =2.0$. (e) $I=45.0$ and $\mu =2.0$. (f) to (h) demonstrate continuous changes of PRCs and membrane potentials ($V$) with $I$ and/or $\mu$ in the class I type ML model. (f) and (g) show PRCs and $V$ at different values of $I$ when $\mu =0.2$ and $=2.0$, respectively. (h) PRCs and $V$ vary with changes of both $I$ and $\mu$. Here the solid lines are the PRC curves while the broken lines are for membrane potential dynamics.

Figure 6

PRC ($Z_V$) shape shifts in a range of $\mu =0.1$ to $2.0$. Red and blue represent, respectively, positive and negative values of $Z_V$. (a) $I=39.7$, (b) $I=40.0$, (c) $I=42.0$, and (d) $I=45.0$.

Figure 7

Linear stability analysis for synchronous states between two neurons. $H_1\left(\phi \right)$, $H_2(-\phi )$, and $\Gamma \left(\phi \right)$ in Eq. (7) are calculated with parameters of $\alpha =1/{\tau }_{\text{syn}}=1.0$ and $I=45.0$. (a) $\mu =2.0$ and (b) $\mu =0.1$. Filled circles designate stable solutions while open circles designate unstable solutions. Two arrows show convergence directions to stable stationary synchronous states, which are demonstrated on numerical simulations in (c) and (d).

Figure 8

$(I,\mu )$-modulated change of the bifurcation diagram for phase differences $\phi$ in the range of $\alpha =(0,1]$. Here $I=39.7$, $40.0$, $42.0$, and $45.0$ while $\mu =2.0$, $1.0$, $0.5$, $0.1$, $0.05$, and $0.01$. Red and green lines, respectively, show that the phase differences are in the unstable and stable states.

Figure 9

A bifurcation diagram for $\phi$ in $\mu =(0,2]$. We set up with $\alpha =1.0$ and $I=45.0$.

Figure 10

Synchronization with or without sine modulations of $\mu$. (a) Simultaneous firings, not accompanying the $\mu$ sine wave. Here $f_1=2$ and $h_1=0.02$. (b) Simultaneous firings tuned with the sine wave, where $h_1=0.08$ for $f_1=2$. (c) Synchronization occurring every two periods for $f_1=4$ and $h_1=0.12$.

Figure 11

Synchronization with sine-wave modulations of $\mu$. (a) $h_1$-$I$ diagrams for synchronization states. (b) Synchronization at different values of $h_1$ after switching the sine-wave amplitude higher.