The rapid transients in spiking models without noise or with slow noise have been reported by several researchers, probably first by Knight (264) and have later been rediscovered several times (183; 77; 351).

The analysis of transients in the escape-rate has been performed in Gerstner (183) where also the limits of high noise and low noise are discussed. For the linearization of the membrane potential density equations and analysis of transient behavior in neuron models with diffusive noise see Brunel and Hakim (78); Lindner and Schimansky-Geier (298); Fourcaud and Brunel (155); Richardson (434). Experimental data on transients in the linearized regime can be found in Silberberg et al. (477); Tchumatchenko et al. (509).

Simoncelli et al. (478) gives an authoritative summary of LNP models. How generalized integrate-and-fire models can be mapped to LNP models has been discussed in Aviel and Gerstner (33) and Ostojic and Brunel (373). An excellent overview of the central concepts of rate models with adaptation can be found in Benda and Herz (48).

1. 1.

   Population of noise-free neurons

   (i) Show that for noise-free neurons the population activity equation yields

   $$A(t)={1\over 1+T^{\prime}(\hat{t})}\,A(\hat{t})\,,$$

   (15.28)

   where $T(\hat{t})$ is the inter-spike interval of a neuron that has fired its last spike at time $\hat{t}$ , and the prime denotes the derivative.

   Hints: In the limit of no noise, the input-dependent interval distribution $P_{I}(t\,|\,\hat{t})$ reduces to a Dirac $\delta$ -function, i.e.,

   $$P_{I}(t\,|\,\hat{t})=\delta[t-\hat{t}-T(\hat{t})]\,.$$

   (15.29)

   where $T(\hat{t})$ is given implicitly by the threshold condition

   $$T(\hat{t})={\rm min}\{(t-\hat{t})\,|\,u(t)=\vartheta;\dot{u}>0,\,t>\hat{t}\}\,.$$

   (15.30)

   Recall from the rules for $\delta$ -functions that

   $$\int_{a}^{b}\delta[f(x)]\,g(x)\,{\text{d}}x={g(x_{0})\over|f^{\prime}(x_{0})|}\,,$$

   (15.31)

   if $f$ has a single zero-crossing $f(x_{0})=0$ in the interval $a<x_{0}<b$ with $f^{\prime}(x_{0})\neq 0$ .

   (ii) Assume SRM ${}_{0}$ neurons with $u(t)=\eta(t-\hat{t})+h(t)$ . Show that

   $$A(t)={h^{\prime}(t)\over\eta^{\prime}T)}\,A(\hat{t})\,,$$

   (15.32)

   Hint: use the results from (i).

   (iii) An input current of amplitue $I_{1}$ is switched on at time $t=0$ . Assume an input potential $h(t)=h_{0}$ for $t<0$ and $h(t)=(R/\tau)\int_{0}^{t}\exp(-s/\tau)I_{1}$ for $t>0$ . Show that the transient of the population activity after the step at $t=0$ is instantaneous, despite the fact the input potential responds slowly.

2. 2.

   LNP and frequency-current curve . Around Eq. ( 15.22 ), it was argued that a model

   $$A(t)=\hat{F}(\int_{0}^{\infty}G_{I}(s)I(t-s){\text{d}}s)\,$$

   (15.33)

   with a choice $\hat{F}(x)=g_{\sigma}(x/[\int_{0}^{\infty}G_{I}(s){\text{d}}s])$ is optimal. The aim is to make the notion of optimality more precise.

   (i) Show that for constant, but arbitrary, input $I_{0}$ , Eq. ( 15.33 ) leads to $A_{0}=g_{\sigma}(I_{0})$ , consistent with the general results of Chapter 12 .

   (ii) Suppose that $G_{I}(s)$ is the linearization of the population activity equations around $A_{0}$ which is achieved for a constant input $I_{0}$ . Show that linearization of Eq. ( 15.33 ) leads to $\Delta A(t)=\int_{0}^{\infty}G(s)\,\Delta I(t-s)\,{\text{d}}s$ .

   Hint: Recall that the response at zero frequency $\hat{G}(0)=\int_{0}^{\infty}G(s){\text{d}}s$ is related to the slope of the gain function.

   (iii) Interpret the results from (i) and (ii) and explain the range of validity of the model defined in Eq. ( 15.33 ). What can happen if the input varies about a mean $I_{1}\neq I_{0}$ ? What happens if the variations around $I_{0}$ are big?

3. 3.

   Leaky integrate-and-fire with white diffusive noise According to the results given in Tab.  15.2 , the linear filter $G_{I}(s)$ of leaky integrate-and-fire neurons has a high-frequency behavior $\tilde{G}_{I}(\omega)=A_{0}{R\over\sigma}{1\over\sqrt{\omega\tau_{m}}}$ .

   (i) Calculate the response to a step current input.

   Hint: Use $\Delta A(t)=\int_{0}^{\infty}G(s)\,\Delta I(t-s)\,{\text{d}}s$ . Insert the step current, take the Fourier transform, perform the multiplication in frequency space, and finish with the inverse Fourier transform.

   (ii) Compare with the simulation results in Fig. 15.9 B.

4. 4.

   Rate model for a population of exponential integrate-and-fire with white diffusive noise

   The aim is to derive the effective time constant given in Eq. ( 15.24 ) which characterizes a population of exponential integrate-and-fire neurons.

   (i) Write $A(t)=F[h(t)]$ . Linearize about a reference value $A_{0}=F(h_{0})$ and prove that ${\text{d}}A/{\text{d}}t=F^{\prime}\,{\text{d}}h/{\text{d}}t$ .

   (ii) Assume that Eq. ( 15.23 ) holds with the unknown time constant $\tau_{\rm eff}$ . Assume periodic stimulation $I(t)=I_{0}+\Delta I\exp(i\omega t)$ with a high frequency $\omega$ . This will lead to a periodic perturbation $\Delta A\,\exp[i(\omega t+\phi)]$ . Find the ratio $c(\omega)=\Delta A/\Delta I$ .

   (iii) Match the high frequency behavior of $c(\omega)$ to $\tilde{G}_{I}(\omega)$ so as to find the time constant $\tau_{\rm eff}$ .

   Hint: Recall from Tab.  15.2 that the linear filter $G_{I}(s)$ of the exponential integrate-and-fire neurons has a high-frequency behavior $\tilde{G}_{I}(\omega)=A_{0}{R\over\Delta_{T}}{1\over\omega\tau_{m}}$ .

5. 5.

   Equivalence of rate models . We use the rate model defined in Eqs. ( 15.1 ) and ( 15.3 ) with $R=1$ in order to describe coupled populations

   $$\tau_{m}{{\text{d}}h_{i}\over{\text{d}}t}=-h_{i}+I_{i}+\sum_{k}w_{ik}F(h_{k})\,.$$

   (15.34)

   Compare this model to another rate model

   $$\tau_{m}{{\text{d}}A_{i}\over{\text{d}}t}=-A_{i}+F(\sum_{k}w_{ik}A_{k}+\hat{I}_{i})\,.$$

   (15.35)

   Show that Eq. ( 15.35 ) implies Eq. ( 15.34 ) under the assumption that $I=\hat{I}+\tau_{m}d\hat{I}/dt$ .

   Hint: Set $h_{i}=\sum_{k}w_{ik}A_{k}+I$ and take the derivative ( 345 ) .