14 The Integral-equation Approach

14.3 Linear response to time-dependent input

We consider a homogeneous population of independent neurons. All neurons receive the same time-dependent input current I(t)I(t) which varies about the mean I0I_{0}. For constant input I0I_{0} the population would fire at an activity A0A_{0} which we can derive from the neuronal gain function. We require that the variations of the input

I(t)=I0+ΔI(t)I(t)=I_{0}+\Delta I(t)\, (14.41)

are small enough for the population activity to stay close to the value A0A_{0}

A(t)=A0+ΔA(t),A(t)=A_{0}+\Delta A(t)\,, (14.42)

with |ΔA|A0|\Delta A|\ll A_{0}.

In that case, we may expand the right-hand side of the population equation A(t)=-tPI(t|t^)A(t^)dt^A(t)=\int_{-\infty}^{t}P_{I}(t|\hat{t})\,A(\hat{t})\,{\text{d}}\hat{t} into a Taylor series about A0A_{0} to linear order in ΔA\Delta A. In this section, we want to show that for spiking neuron models (either integrate-and-fire or SRM0{}_{0} neurons) the linearized population equation can be written in the form

ΔA(t)=-tP0(t-t^)ΔA(t^)dt^+A0ddt0(x)Δh(t-x)dx,\Delta A(t)=\int_{-\infty}^{t}P_{0}(t-\hat{t})\,\Delta A(\hat{t})\,{\text{d}}% \hat{t}+A_{0}\,{{\text{d}}\over{\text{d}}t}\int_{0}^{\infty}{\mathcal{L}}(x)\,% \Delta h(t-x)\,{\text{d}}x\,, (14.43)

where P0(t-t^)P_{0}(t-\hat{t}) is the interval distribution for constant input I0I_{0}, (x){\mathcal{L}}(x) is a real-valued function that plays the role of an integral kernel, and

Δh(t)=0κ(s)ΔI(t-s)ds\Delta h(t)=\int_{0}^{\infty}\kappa(s)\,\Delta I(t-s)\,{\text{d}}s (14.44)

is the input potential generated by the time-dependent part of the input current. The first term of the right-hand side of Eq. (14.43) takes into account that previous perturbations ΔA(t^)\Delta A(\hat{t}) with t^<t\hat{t}<t have an after-effect one inter-spike interval later. The second term describes the immediate response to a change in the input potential. If we want to understand the response of the population to an input current ΔI(t)\Delta I(t), we need to know the characteristics of the kernel (x){\mathcal{L}}(x). The main task of this section is therefore the calculation of (x){\mathcal{L}}(x) to be performed in Section 14.3.1.

The linearization of the integral equation (183) is analogous to the linearization of the membrane potential density equations (78) that was presented in Chapter 13. In order to arrive at the standard formula for the linear response,

A(t)=A0+0G(s)ΔI(t-s)dsA(t)=A_{0}+\int_{0}^{\infty}G(s)\,\Delta I(t-s)\,{\text{d}}s (14.45)

we insert Eq. (14.44) into Eq. (14.43) and take the Fourier transform. For ω0\omega\neq 0 we find

A^(ω)=iωA0^(ω)κ^(ω)1-P^(ω)I^(ω)=G^(ω)I^(ω).\hat{A}(\omega)={i\omega}\,{A_{0}\,\hat{{\mathcal{L}}}(\omega)\,\hat{\kappa}(% \omega)\over 1-\hat{P}(\omega)\,}\,\hat{I}(\omega)=\hat{G}(\omega)\,\hat{I}(% \omega)\,. (14.46)

Hats denote transformed quantities, i.e., κ^(ω)=κ0(s)exp(-iωs)ds\hat{\kappa}(\omega)=\int\kappa_{0}(s)\,\exp(-i\,\omega\,s)\,{\text{d}}s is the Fourier transform of the response kernel; P^(ω)\hat{P}(\omega) is the Fourier transform of the interval distribution P0(t-t^)P_{0}(t-\hat{t}); and ^(ω)\hat{{\mathcal{L}}}(\omega) is the transform of the kernel {\mathcal{L}}. Note that for ω0\omega\neq 0 we have A^(ω)=(ΔA)^(ω)\hat{A}(\omega)=\hat{(\Delta A)}(\omega) and I(ω)=(ΔI^)(ω)I(\omega)=\hat{(\Delta I})(\omega) since A0A_{0} and I0I_{0} are constant.

The frequency dependent gain G^(ω)\hat{G}(\omega) describes the linear response of the population activity to a periodic input current I^(ω)\hat{I}(\omega). The linear response filter G(s)G(s) in the time-domain is found by inverse Fourier transform

G(s)=12π-G^(ω)e+iωsdω.G(s)={1\over 2\pi}\int_{-\infty}^{\infty}\hat{G}(\omega)\,e^{+i\,\omega\,s}{% \text{d}}\omega\,. (14.47)

A0A_{0} is the mean rate for constant drive I0I_{0}. The filter GG plays an important role for the analysis of the stability of the stationary state in recurrent networks (Section 14.2).

Example: Leaky integrate-and-fire with escape noise

The frequency-dependent gain G^\hat{G} depends on the filter (s){\mathcal{L}}(s) which in turns depends on the width of the interval distribution P0(s)P_{0}(s) (Fig. 14.9B and A, respectively). In Fig. 14.9C we have plotted the signal gain G^(ω)\hat{G}(\omega) for integrate-and-fire neurons with escape noise at different noise levels. At low noise, the signal gain exhibits resonances at the frequency that corresponds to the inverse of the mean interval and multiples thereof. Increasing the noise level, however, lowers the signal gain of the system. For high noise (long-dashed line in Fig. 14.9C) the signal gain at 1000 Hz is ten times lower than the gain at zero frequency. The cut-off frequency depends on the noise level. The gain at zero frequency corresponds to the slope of the gain function gσ(I0)g_{\sigma}(I_{0}) and changes with the level of noise.

Fig. 14.9: Response properties of a population of leaky integrate-and-fire neurons with escape noise. A. Interval distribution for three different noise levels. The escape rate has been taken as piecewise linear ρ=ρ0[u-ϑ](u-ϑ)\rho=\rho_{0}\,[u-\vartheta]{\mathcal{H}}(u-\vartheta). The value of the bias current I0I_{0} has been adjusted so that the mean interval is always T\langle T\rangle =8 ms. B. The corresponding kernel IF(x){\mathcal{L}}^{\rm IF}(x). The dip in the kernel around x=Tx=\langle T\rangle is typical for integrate-and-fire neurons. C. Frequency dependent gain |G(f)|=|G^(ω=2πf)||G(f)|=|\hat{G}(\omega=2\pi f)|. Low noise (short dashed line): the sharply peaked interval distribution (standard deviation 0.750.75 ms) and rapid fall-off of kernel {\mathcal{L}} lead to a linear response gain G^\hat{G} with strong resonances at multiples of the intrinsic firing frequency 1/T1/\langle T\rangle. High noise (long-dashed line): the broad interval distribution (standard deviation 44 ms) and broad kernel {\mathcal{L}} suppress resonances in the frequency dependent gain. Medium noise (solid line): A single resonance for an interval distribution with standard deviation 22 ms. Adapted from (183).

14.3.1 Derivation of the linear response filter (*)

In order to derive the linearized response ΔA\Delta A of the population activity to a change in the input we start from the conservation law,

1=-tSI(t|t^)A(t^)dt^,1=\int_{-\infty}^{t}S_{I}(t\,|\,\hat{t})\,A(\hat{t})\,{\text{d}}\hat{t}\,, (14.48)

cf. Eq. (14.8). As we have seen in Section 14.1 the population equation (14.5) can be obtained by taking the derivative of Eq. (14.8) with respect to tt, i.e.,

0=ddt-tSI(t|t^)A(t^)dt^.0={{\text{d}}\over{\text{d}}t}\int_{-\infty}^{t}S_{I}(t\,|\,\hat{t})\,A(\hat{t% })\,{\text{d}}\hat{t}\,. (14.49)

For constant input I0I_{0}, the population activity has a constant value A0A_{0}. We consider a small perturbation of the stationary state, A(t)=A0+ΔA(t)A(t)=A_{0}+\Delta A(t), that is caused by a small change in the input current, ΔI(t)\Delta I(t). The time-dependent input generates a total postsynaptic potential, h(t)=h0+Δh(t,t^)h(t)=h_{0}+\Delta h(t,\hat{t}) where h0h_{0} is the postsynaptic potential for constant input I0I_{0} and

Δh(t,t^)=0b(t^)κ(s)ΔI(t-s)ds\Delta h(t,\hat{t})=\int_{0}^{b(\hat{t})}\kappa(s)\,\Delta I(t-s)\,{\text{d}}s (14.50)

is the change of the postsynaptic potential generated by ΔI\Delta I. Note that we keep the notation general and include a dependence upon the last firing time t^\hat{t}. For leaky integrate-and-fire neurons, we set b(t^)=t-t^b(\hat{t})=t-\hat{t} whereas for SRM0{}_{0} neurons we set b(t^)=b(\hat{t})=\infty. We expand Eq. (14.49) to linear order in ΔA\Delta A and Δh\Delta h and find

0=ddt-tS0(t-t^)ΔA(t^)dt^+A0ddt{-tds-tdt^Δh(s,t^)SI(t|t^)Δh(s,t^)|Δh=0}.0={{\text{d}}\over{\text{d}}t}\int_{-\infty}^{t}S_{0}(t-\hat{t})\,\Delta A(% \hat{t})\,{\text{d}}\hat{t}\\ +A_{0}\,{{\text{d}}\over{\text{d}}t}\left\{\int_{-\infty}^{t}{\text{d}}s\,\int% _{-\infty}^{t}{\text{d}}\hat{t}\,\Delta h(s,\hat{t})\,\left.{\partial S_{I}(t% \,|\,\hat{t})\over\partial\Delta h(s,\hat{t})}\right|_{\Delta h=0}\right\}\,. (14.51)

We have used the notation S0(t-t^)=SI0(t|t^)S_{0}(t-\hat{t})=S_{I_{0}}(t\,|\,\hat{t}) for the survivor function of the asynchronous firing state. To take the derivative of the first term in Eq. (14.51) we use dS0(s)/ds=-P0(s){{\text{d}}}S_{0}(s)/{\text{d}}s=-P_{0}(s) and S0(0)=1S_{0}(0)=1. This yields

ΔA(t)=-tP0(t-t^)ΔA(t^)dt^-A0ddt{-tds-tdt^Δh(s,t^)SI(t|t^)Δh(s,t^)|Δh=0}.\Delta A(t)=\int_{-\infty}^{t}P_{0}(t-\hat{t})\,\Delta A(\hat{t})\,{\text{d}}% \hat{t}\\ \mbox{}-A_{0}\,{{\text{d}}\over{\text{d}}t}\left\{\int_{-\infty}^{t}{\text{d}}% s\,\int_{-\infty}^{t}{\text{d}}\hat{t}\,\Delta h(s,\hat{t})\,\left.{\partial S% _{I}(t\,|\,\hat{t})\over\partial\Delta h(s,\hat{t})}\right|_{\Delta h=0}\right% \}\,. (14.52)

We note that the first term on the right-hand side of Eq. (14.52) has the same form as the population integral equation (14.5), except that P0P_{0} is the interval distribution in the stationary state of asynchronous firing.

To make some progress in the treatment of the second term on the right-hand side of Eq. (14.52), we now restrict the choice of neuron model and focus on either SRM0{}_{0} or integrate-and-fire neurons.

(i) For SRM0{}_{0} neurons, we may drop the t^\hat{t} dependence of the potential and set Δh(t,t^)=Δh(t)\Delta h(t,\hat{t})=\Delta h(t) where Δh\Delta h is the input potential caused by the time-dependent current ΔI\Delta I; compare Eqs. (14.44) and (14.50). This allows us to pull the variable Δh(s)\Delta h(s) in front of the integral over t^\hat{t} and write Eq. (14.52) in the form

ΔA(t)=-tP0(t-t^)ΔA(t^)dt^+A0ddt0(x)Δh(t-x)dx.\Delta A(t)=\int_{-\infty}^{t}P_{0}(t-\hat{t})\,\Delta A(\hat{t})\,{\text{d}}% \hat{t}+A_{0}\,{{\text{d}}\over{\text{d}}t}\int_{0}^{\infty}{\mathcal{L}}(x)\,% \Delta h(t-x)\,{\text{d}}x\,. (14.53)

with a kernel

(x)=-xdξS(ξ|0)Δh(ξ-x)SRM(x).{\mathcal{L}}(x)=-\int_{x}^{\infty}d\xi\,{\partial S(\xi|0)\over\partial\Delta h% (\xi-x)}\equiv{\mathcal{L}}^{\rm SRM}(x)\,. (14.54)

(ii) For leaky integrate-and-fire neurons we set Δh(t,t^)=Δh(t)-Δh(t^)exp[-(t-t^)/τ]\Delta h(t,\hat{t})=\Delta h(t)-\Delta h(\hat{t})\,\exp[-(t-\hat{t})/\tau], because of the reinitialization of the membrane potential after the reset (183). After some rearrangements of the terms, Eq. (14.52) becomes identical to Eq. (14.53) with a kernel

(x)=-xdξS(ξ|0)Δh(ξ-x)+0xdξe-ξ/τS(x|0)Δh(ξ)IF(x).{\mathcal{L}}(x)=-\int_{x}^{\infty}d\xi\,{\partial S(\xi|0)\over\partial\Delta h% (\xi-x)}+\int_{0}^{x}d\xi\,e^{-\xi/\tau}\,{\partial S(x|0)\over\partial\Delta h% (\xi)}\equiv{\mathcal{L}}^{\rm IF}(x)\,. (14.55)

Let us discuss Eq. (14.53). The first term on the right-hand side of Eq. (14.53) is of the same form as the dynamic equation (14.5) and describes how perturbations ΔA(t^)\Delta A(\hat{t}) in the past influence the present activity ΔA(t)\Delta A(t). The second term gives an additional contribution which is proportional to the derivative of a filtered version of the potential Δh\Delta h.

We see from Fig. 14.10 that the width of the kernel {\mathcal{L}} depends on the noise level. For low noise, it is significantly sharper than for high noise.

Fig. 14.10: Interval distribution (A) and the kernel SRM(x){\mathcal{L}}^{\rm SRM}(x) (B) for SRM0{}_{0} neurons with escape noise. The escape rate has been taken as piecewise linear ρ=ρ0[u-ϑ](u-ϑ)\rho=\rho_{0}\,[u-\vartheta]{\mathcal{H}}(u-\vartheta). For low noise (solid lines in A and B) the interval distribution is sharply peaked and the kernel SRM{\mathcal{L}}^{\rm SRM} has a small width. For high noise (dashed line) both the interval distribution and the kernel SRM{\mathcal{L}}^{\rm SRM} are broad. The value of the bias current I0I_{0} has been adjusted so that the mean interval is always 40 ms. The kernel has been normalized to 0(x)dx=1\int_{0}^{\infty}{\mathcal{L}}(x)\,{\text{d}}x=1.

Example: The kernel (x){\mathcal{L}}(x) for escape noise (*)

In the escape noise model, the survivor function is given by

SI(t|t^)=exp{-t^tf[η(t-t^)+h(t,t^)]   d   t}S_{I}(t\,|\,\hat{t})=\exp\left\{-\int_{\hat{t}}^{t}f[\eta(t^{\prime}-\hat{t})+% h(t^{\prime},\hat{t})]\,{\text{d}}t^{\prime}\right\} (14.56)

where f[u]f[u] is the instantaneous escape rate across the noisy threshold; cf. Chapter 7. We write h(t,t^)=h0(t-t^)+Δh(t,t^)h(t,\hat{t})=h_{0}(t-\hat{t})+\Delta h(t,\hat{t}). Taking the derivative with respect to Δh\Delta h yields

SI(t|t^)Δh(s,t^)|Δh=0=-(s-t^)(t-s)f[η(s-t^)+h0(s-t^)]S0(t-t^)\left.{\partial S_{I}(t\,|\,\hat{t})\over\partial\Delta h(s,\hat{t})}\right|_{% \Delta h=0}=-{\mathcal{H}}(s-\hat{t})\,{\mathcal{H}}(t-s)\,f^{\prime}[\eta(s-% \hat{t})+h_{0}(s-\hat{t})]\,S_{0}(t-\hat{t}) (14.57)

where S0(t-t^)=Sh0(t|t^)S_{0}(t-\hat{t})=S_{h_{0}}(t\,|\,\hat{t}) and f=df(u)/duf^{\prime}={\text{d}}f(u)/{\text{d}}u. For SRM0{}_{0}-neurons, we have h0(t-t^)h0h_{0}(t-\hat{t})\equiv h_{0} and Δh(t,t^)=Δh(t)\Delta h(t,\hat{t})=\Delta h(t), independent of t^\hat{t}. The kernel {\mathcal{L}} is therefore

SRM(t-s)=(t-s)-s   d   t^f[η(s-t^)+h0]S0(t-t^).{\mathcal{L}}^{\rm SRM}(t-s)={\mathcal{H}}(t-s)\int_{-\infty}^{s}{\text{d}}% \hat{t}\,f^{\prime}[\eta(s-\hat{t})+h_{0}]\,S_{0}(t-\hat{t})\,. (14.58)

Example: Absolute refractoriness (*)

Absolute refractoriness is defined by a refractory kernel η(s)=-\eta(s)=-\infty for 0<s<δabs0<s<\delta^{\rm abs} and zero otherwise. We take an arbitrary escape rate f(u)0f(u)\geq 0. The only condition on ff is that the escape rate goes rapidly to zero for voltages far below threshold: limu-f(u)=0=limu-f(u){\rm lim}_{u\to-\infty}f(u)=0={\rm lim}_{u\to-\infty}f^{\prime}(u).

This yields f[η(t-t^)+h0]=f(h0)(t-t^-δabs)f[\eta(t-\hat{t})+h_{0}]=f(h_{0})\,{\mathcal{H}}(t-\hat{t}-\delta^{\rm abs}) and hence

f[η(t-t^)+h0]=f(h0)(t-t^-δabs).f^{\prime}[\eta(t-\hat{t})+h_{0}]=f^{\prime}(h_{0})\,{\mathcal{H}}(t-\hat{t}-% \delta^{\rm abs})\,. (14.59)

The survivor function S0(s)S_{0}(s) is unity for s<δabss<\delta^{\rm abs} and decays as exp[-f(h0)(s-δabs)]\exp[-f(h_{0})\,(s-\delta^{\rm abs})] for s>δabss>\delta^{\rm abs}. Integration of Eq. (14.58) yields

(t-t1)=(t-t1)f(h0)f(h0)exp[-f(h0)(t-t1)].{\mathcal{L}}(t-t_{1})={\mathcal{H}}(t-t_{1}){f^{\prime}(h_{0})\over f(h_{0})}% \exp[-f(h_{0})\,(t-t_{1})]\,. (14.60)

As we have seen in Section 14.1, absolute refractoriness leads to the Wilson-Cowan integral equation (14.10). Thus {\mathcal{L}} defined in (14.60) is the kernel relating to Eq. (14.10). It could have been derived directly from the linearization of the Wilson-Cowan integral equation (see Exercises). We note that it is a low-pass filter with cut-off frequency f(h0)f(h_{0}), which depends on the input potential h0h_{0}.