Impact of membrane bistability on dynamical response of neuronal populations

Neurons in many brain areas can develop a pronounced depolarized state of membrane potential (up state) in addition to the normal hyperpolarized state near the resting potential. The influence of the up state on signal encoding, however, is not well investigated. Here we construct a one-dimensional bistable neuron model and calculate the linear dynamical response to noisy oscillatory inputs analytically. We find that with the appearance of an up state, the transmission function is enhanced by the emergence of a local maximum at some optimal frequency and the phase lag relative to the input signal is reduced. We characterize the dependence of the enhancement of frequency response on intrinsic dynamics and on the occupancy of the up state.

Figure 1

Illustration of the model. (a): illustration of the piecewise linear model; (b): MP trajectories for $r_1=1$, 5, and 10 from bottom to top. Parameters used are: $r=-1,v_0=0.5,v_r=v_{t1},v_{t0}=2,{\tilde{v}}_b=-0.2,\tau =10\text{ms},{\tau }_r=0\text{ms},\mu =0,\sigma =0.5$.

Figure 2

Dependence of stationary probability density and linear dynamical response on $r_1$ and $r$. (a) and (b): probability density for different $r_1$ and $r$. (c) and (d): dependence of the transmission function (upper panels) and phase lag (lower panels) of the linear dynamical response on $r_1$ and $r$. The linear dynamical response is normalized with the value at $f=1\text{Hz}$. Signal frequency $f$ is related to the angular frequency $\omega$ by $\omega =2\pi f$. Solid lines are from theoretical results and asterisks are from simulations. Parameters used: $r=-1$ in (a) and (c), $r_1=10$ in (b) and (d). Other parameters are the same as in Fig. 1.

Figure 3

Dependence of up state occupancy and resonance in the linear dynamical response on $r_1$ and $r$. (a) and (b): the up state occupancy $P_0^{\mathrm{up}}/P_0^{\mathrm{down}}$ as a function of $r_1$ (a) and $\left|r\right|$ (b). (c) and (d): the local maximal value ${\nu }_{1c}^{max}$ of the normalized transmission function as a function of $r_1$ (c) and $\left|r\right|$ (d). Insets: ${\nu }_{1c}^{max}$ as a function of $P_0^{\mathrm{up}}/P_0^{\mathrm{down}}$ due to the change of $r_1$ (c) and $\left|r\right|$ (d). (e) and (f): the resonance frequency $f^{max}$ as a function of $r_1$ (e) and $\left|r\right|$ (f). Parameters used: $r=-1$ in (a) and $r_1=10$ in (b). Other parameters are the same as in Fig. 1.

Figure 4

Impact of bistability on linear dynamical response in a biophysical realistic model. (a): membrane dynamics on the nullcline of activation variable $m$ ($m=m_{\infty }$) with $g_{KS}=0\text{and}5$. Here $\dot{V}$ in the vertical axis is the time derivative of $V$ obtained by taking ${\mu }_{\mathrm{ext}}=0$ and $\sigma =0$ in Eq. (C1). (b): trajectory of membrane potential for $g_{KS}=5$ and $\sigma =10\text{mV}$. (c): normalized transmission functions for the two different $g_{KS}$. Note that ${\mu }_{\mathrm{ext}}$ is reduced to $-4\text{mV}$ when $g_{KS}=0$ to have similar stationary firing rates in the two cases (about 10 Hz).