Stein’s neuronal model with pooled renewal input

Abstract

The input of Stein’s model of a single neuron is usually described by using a Poisson process, which is assumed to represent the behaviour of spikes pooled from a large number of presynaptic spike trains. However, such a description of the input is not always appropriate as the variability cannot be separated from the intensity. Therefore, we create and study Stein’s model with a more general input, a sum of equilibrium renewal processes. The mean and variance of the membrane potential are derived for this model. Using these formulas and numerical simulations, the model is analyzed to study the influence of the input variability on the properties of the membrane potential and the output spike trains. The generalized Stein’s model is compared with the original Stein’s model with Poissonian input using the relative difference of variances of membrane potential at steady state and the integral square error of output interspike intervals. Both of the criteria show large differences between the models for input with high variability.

Appendix

Here we derive formulas for mean and variance of $S(t)$ and $S$. It is clearly sufficient to derive mean and variance of quantity

$$\begin{aligned} \tilde{S}(t) = \sum _{i=1}^{N(t)} A_i \mathrm{e}^{-\frac{t-X_i}{\tau }}, \end{aligned}$$

(25)

where $N(t)$ and $X_i,\,i=1,\ldots ,N(t),$ correspond to an equilibrium renewal process with intensity $\lambda$ and $A_i,\,i=1,\ldots ,N(t),$ are positive independent and identically distributed random variables with mean $\mu$ and variance $\sigma ^2$. The formulas (16), (17), (18) and (19) will be simple consequences.

Firstly, let us denote,

$$\begin{aligned} \tilde{S}_0(t) = \sum _{i=1}^{N(t)} \mathrm{e}^{-\frac{t-X_i}{\tau }} \end{aligned}$$

(26)

and have $\Delta t > 0$ and points $t_i = i\Delta t,\,i=0,\ldots ,\lfloor t/\Delta t\rfloor$. The number of events in interval $[t_{i-1},t_i)$ is denoted as $N_i,$ $i=1,\ldots ,\lfloor t/\Delta t\rfloor$. Then

$$\begin{aligned} \tilde{S}_0(t) = \lim _{\Delta t \rightarrow 0}\sum ^{\lfloor t/\Delta t\rfloor }_{i=1}N_i \mathrm{e}^{-\frac{t-t_i}{\tau }}, \end{aligned}$$

(27)

and thus

$$\begin{aligned} \mathrm {E}(\tilde{S}_0(t))= & {} \lim _{\Delta t \rightarrow 0}\!\sum ^{\lfloor t/\Delta t\rfloor }_{i=1}\mathrm {E}(N_i) \mathrm{e}^{-\frac{t-t_i}{\tau }} \!=\! \lim _{\Delta t \rightarrow 0}\!\sum ^{\lfloor t/\Delta t\rfloor }_{i=1}\lambda \Delta t \mathrm{e}^{-\frac{t-t_i}{\tau }}\nonumber \\= & {} \lambda \int _0^t \mathrm{e}^{-\frac{t-x}{\tau }}\,\mathrm {d}x= \lambda \tau \left( 1-\mathrm{e}^{-\frac{t}{\tau }}\right) , \end{aligned}$$

(28)

which simply yields

$$\begin{aligned} \mathrm {E}(\tilde{S}(t)) = \mu \lambda \tau \left( 1-\mathrm{e}^{-\frac{t}{\tau }}\right) . \end{aligned}$$

(29)

Before deriving the formulas for variance, a necessary relationship is presented. It holds (Cox and Lewis 1966)

$$\begin{aligned} \lim _{\Delta t \rightarrow 0} \frac{\,\mathrm {Cov}(\Delta N_t,\Delta N_{t+x})}{\Delta t^2} = \lambda (\lambda _o(x)-\lambda +\delta (x)), \end{aligned}$$

(30)

where $\Delta N_t$ denotes the number of events in interval $[t,t+\Delta t]$ and $\delta (x)$ is the Dirac delta function. Note that in neuronal modeling the function (30) is known as correlation function of the spike train (Gerstner and Kistler 2002). Then

$$\begin{aligned} \mathrm {Var}(\tilde{S}_0(t))= & {} \lim _{\Delta t \rightarrow 0}\sum ^{\lfloor t/\Delta t\rfloor }_{i=1}\sum ^{\lfloor t/\Delta t\rfloor }_{j=1} \mathrm{e}^{-\frac{t-t_i}{\tau }} \mathrm{e}^{-\frac{t-t_j}{\tau }} \,\mathrm {Cov}(N_i,N_j) \nonumber \\= & {} \lim _{\Delta t\rightarrow 0}\!\sum ^{\lfloor t/\Delta t\rfloor }_{i=1}\! \Delta t\, \mathrm{e}^{-\frac{t-t_i}{\tau }} \sum ^{\lfloor t/\Delta t\rfloor }_{j=1} \Delta t\, \mathrm{e}^{-\frac{t-t_j}{\tau }}\frac{\,\mathrm {Cov}(N_i,N_j)}{\Delta t^2} \nonumber \\= & {} \lambda \int _{0}^t\int _{0}^t \mathrm{e}^{-\frac{x+y}{\tau }}(\lambda _o(|y-x|)-\lambda )\,\mathrm {d}x\,\mathrm {d}y\nonumber \\&+\, \frac{1}{2}\lambda \tau (1-\mathrm{e}^{-\frac{2t}{\tau }}). \end{aligned}$$

(31)

Next,

$$\begin{aligned} \mathrm {Var}(\tilde{S}(t))= & {} \mathrm {Var}\left( \sum _{i=1}^{N(t)} A_i \mathrm{e}^{-\frac{t-X_i}{\tau }}\right) = \sum _{i=1}^{N(t)} \mathrm {Var}\left( A_i\mathrm{e}^{-\frac{t-X_i}{\tau }}\right) \nonumber \\&+\, \sum _{i=1}^{N(t)} \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}} ^{N(t)} \,\mathrm {Cov}\left( A_i\mathrm{e}^{-\frac{t-X_i}{\tau }},A_j\mathrm{e}^{-\frac{t-X_j}{\tau }}\right) \nonumber \\= & {} \mu ^2\mathrm {Var}(\tilde{S}_0(t))+\sigma ^2\mathrm {E}\left( \sum _{i=1}^{N(t)}\mathrm{e}^{-2\frac{t-X_i}{\tau }}\right) \nonumber \\= & {} \mu ^2\lambda \int _{0}^t\int _{0}^t \mathrm{e}^{-\frac{x+y}{\tau }}(\lambda _o(|y-x|)-\lambda )\,\mathrm {d}x\,\mathrm {d}y\nonumber \\&+\, \frac{1}{2}\mu ^2\lambda \tau \left( 1-\mathrm{e}^{-\frac{2t}{\tau }}\right) + \frac{1}{2}\sigma ^2\lambda \tau \left( 1-\mathrm{e}^{-\frac{2t}{\tau }}\right) .\nonumber \\ \end{aligned}$$

(32)

Finally, we derive the limit of the previous relationship for $t \rightarrow \infty$. Using formula (5) we obtain

$$\begin{aligned} \lim _{t \rightarrow \infty } \mathrm {Var}(\tilde{S}(t))= & {} 2\mu ^2\lambda \tau \int _{0}^\infty \mathrm{e}^{-\frac{2x}{\tau }}\sum _{k=1}^{\infty }\int _{0}^\infty \mathrm{e}^{-\frac{y}{\tau }}f_k(y)\,\mathrm {d}y\,\mathrm {d}x\nonumber \\&-\, \mu ^2\lambda ^2\tau ^2+ \frac{1}{2}\mu ^2\lambda \tau + \frac{1}{2}\sigma ^2\lambda \tau \nonumber \\= & {} \frac{\mu ^2\lambda \tau }{1-\fancyscript{L}\{f\}(1/\tau )} - \mu ^2\lambda ^2\tau ^2 - \frac{1}{2}\mu ^2\lambda \tau \nonumber \\&+\, \frac{1}{2}\sigma ^2\lambda \tau . \end{aligned}$$

(33)