The faithful copy neuron

Abstract

Theoretical and experimental evidence is presented for the presence in nervous tissue of neurons whose firing rate faithfully follow their input stimulus. Such neurons are shown to deliver their spikes with minimum dissipation per spike. This optimal performance is likely accomplished by use of local circuitry that adjusts conductances to match input currents so that the neuron operates near the threshold for firing. This results in an unusual mechanism for neuronal firing that uses background noise to achieve the desired firing rate. This framework takes place dynamically, and the present deliberations apply under time varying conditions. It is shown that an analytically explicit probability distribution function, which depends on one dimensionless parameter, can account for the interspike interval statistics under general time varying conditions. An innovative analysis based on the unsteady firing rate fits data to the appropriate probability distribution function.

Appendix: General case

Instead of the spike counting clock (Eq. (8)), Eq. (18) is more suitably transformed by

$$\label{eq15a} \tau(t)=\frac{1}{\widehat{s}}\int^t_0s\left(t^\prime\right)dt^\prime. $$

(34)

In addition we write

$$\label{eq16} x=1-\sqrt{\epsilon}z $$

(35)

and

$$\label{eq16a} \beta=(\widehat{s}-1)/\sqrt{\epsilon} $$

(36)

so that Eq. (18) becomes

$$\label{eq17} \frac{\partial \rho}{\partial\tau}=\frac{\partial}{\partial z}\bigg[(z+\beta)\rho+\frac{\partial \rho}{\partial z}\bigg]; 0\leq z\leq \infty, $$

(37)

Sirovich and Knight (2011). The time transformation (Eq. (34)) uses γ(t) as a temporal density in order to achieve the form of Eq. (37). Observe that the interspike interval based on Eq. (34), τ ^I satisfies

$$\label{eq17a} \kappa\widehat{s}\tau^I=\widetilde{\tau}^I, $$

(38)

and an immediate consequence is

$$\label{eq17b} \left\langle\tau^I\right\rangle=1/\left(\widehat{s}\kappa\left(\epsilon,\widehat{s}\right)\right). $$

(39)

In terms of Eq. (37) the initial condition becomes

$$\label{eq18} \rho(z,\tau=0)=\delta\left(z-1/\sqrt{\epsilon}\right) $$

(40)

and the absorbing boundary condition is now at the origin

$$\label{eq19} \rho(0,\tau)=0. $$

(41)

Using standard arguments the interspike interval pdf is given by

$$\label{eq20} {\cal P}\left(\tau^I;\beta, \epsilon\right)=\frac{\partial}{\partial z}\rho\left(z,\tau^I;\beta, \epsilon\right)\big|_{z=0} $$

(42)

where ρ is the solution to the above stated problem. It can be shown (Sirovich and Knight 2011) that ${\cal P}(\tau)$ (Eq. (42)) satisfies

$$\label{eq21} \frac{e^{\left(e^{-\tau^I}/\sqrt{\epsilon}-a\left(\tau^I\right)\right)^2/2v\left(\tau^I\right)}}{\sqrt{2\pi v\left(\tau^I\right)}}=\int^{\tau^I}_0d\sigma\frac{e^{-a(\sigma)^2/2v(\sigma)}}{\sqrt{2\pi v(\sigma)}}P(\tau-\sigma) $$

(43)

where

$$\label{eq22} a(\tau)=\beta\left(1-e^{-\tau}\right) $$

(44)

and

$$\label{eq23} v(\tau)=1-e^{-2\tau} $$

(45)

Equation (43), a convolution Volterra equation, can be solved numerically with ease, and one can easily build up an extensive class of interspike interval pdfs. For data analysis roughly 142,000 pdfs covering − 3 ≤ β < 3,.01 < ε < .6 in increments of 0.005 were constructed. This construction, based on an enhanced trapezoidal rule, led to the high resolution collection Eq. (12) mentioned earlier. The algorithm when tested against the exact result, Eq. (19), showed a fractional error of 0(10^− 5).