A single spiking neuron that can represent interval timing: analysis, plasticity and multi-stability

Abstract

The ability to represent interval timing is crucial for many common behaviors, such as knowing whether to stop when the light turns from green to yellow. Neural representations of interval timing have been reported in the rat primary visual cortex and we have previously presented a computational framework describing how they can be learned by a network of neurons. Recent experimental and theoretical results in entorhinal cortex have shown that single neurons can exhibit persistent activity, previously thought to be generated by a network of neurons. Motivated by these single neuron results, we propose a single spiking neuron model that can learn to compute and represent interval timing. We show that a simple model, reduced analytically to a single dynamical equation, captures the average behavior of the complete high dimensional spiking model very well. Variants of this model can be used to produce bi-stable or multi-stable persistent activity. We also propose a plasticity rule by which this model can learn to represent different intervals and different levels of persistent activity.

Appendices

Appendix

A Calcium dynamics and their analysis

The calcium current in this model arises from high threshold calcium channels, which move to an open state only when the neuron spikes and return to closed rapidly after the end of the action potential (rapidly means much faster that τ _Ca).

We simulate these calcium channels using the simple linear expression, instead of the GHK (Johnston and Wu 1994) equation for simplicity:

$$ I_{\rm Ca}=g_{\rm Ca}(V-E_{\rm Ca}) $$

(19)

where V is the postsynaptic potential, E _Ca the reversal potential for calcium, where g _Ca is governed by the dynamical equation:

$$ \tau_{\rm gca}\dot g_{\rm Ca}=-g_{\rm Ca}+H_{\rm Ca}(V,\theta,m) $$

(20)

where H _Ca describes the voltage dependence of the calcium channels. Typically we use a hill function with a threshold θ and a hill coefficient m. However since τ _gca < < τ _Ca and since voltage changes are rapid as well, the calcium flows in during a very short period, and can be approximated as a step change in calcium levels.

This step is the total amount of calcium transferred through these channels during a single AP, which we define as ρ. The parameter ρ is a function the calcium channel parameters, such as it’s maximal conductance and it’s time time constant. With this definition and the step approximation Eqs. (1 and 19–20) can be replaced by a single equation of the form:

$${d{\rm Ca}\over dt}=-{1\over \tau_{\rm Ca}}{\rm Ca}+\rho\sum\limits_i^n\delta(t-t_i) $$

(21)

where t _i are spike times, and t _n  < t. In our simulations we calculate ρ from simulations of the detailed calcium dynamics by integrating over the total current generated by a single action potential. We find by comparing the simulations with the detailed dynamics and simulations with the step approximation that this approximation leads only to minor quantitative differences.

If the postsynaptic neuron fires at a fixed frequency f with a corresponding inter spike interval Δt = 1/f, so that t _i  = iΔt we get that:

$${\rm Ca}(t)=e^{-t/\tau_{\rm Ca}}\rho{1-a^{n+1}\over 1-a} $$

(22)

where $a=\exp(-\Delta t/\tau_{\rm Ca})$. Defining t ^′ = t − nΔt, the time since the last spike we get that:

$${\rm Ca}\left(t^\prime\right)=e^{-t^\prime\over \tau_{\rm Ca}}\rho {{e^{-n\Delta t\over \tau_{\rm Ca}}-e^{\Delta t\over \tau_{\rm Ca}}} \over 1-e^{\Delta t\over \tau_{\rm Ca}}} $$

(23)

for larger n this is approximated by

$${\rm Ca}\left(t^\prime\right)\approx e^{-t^\prime/ \tau_{\rm Ca}}\rho{{{e^{\Delta t\over \tau_{\rm Ca}}}}\over {1-e^{\Delta t\over \tau_{\rm Ca}}} } $$

(24)

which can be rewritten as:

$${\rm Ca}\left(t^\prime\right)= \exp\left(-t^\prime/ \tau_{\rm Ca}\right){\rm Ca}^0(f) $$

(25)

where, Ca⁰(f) is the value of calcium shortly after a spike.

$${\rm Ca}^0(f) =\rho{\exp(1/f\tau_{\rm Ca})\over {\exp(1/f\tau_{\rm Ca})-1}} $$

(26)

By averaging over a single inter-spike interval we get:

$${\overline{\rm Ca}} = {\rm Ca}^0(f)*\left(1-{\rm exp}(-1/f\tau_{\rm Ca})\right)\tau_{\rm ca}f=\rho\tau_{\rm Ca}\cdot f $$

(27)

If we replace Eq. (21) by the equation:

$${d{\rm Ca}\over dt}=-{1\over \tau_{\rm Ca}}{\rm Ca}+\rho f . $$

(28)

we obtain at steady state the same expression as in Eq. (27). Equation (28) is the same as Eq. (3).

B Parameters

Note the parameter ρ is calculated inside the code by integrating the calcium dynamics for a single postsynaptic spike and finding the max of the calcium transient. It depends on the parameters V _theta, E _Ca, m, and also on the properties of the action potential. Here we assume that action potential are exponentials with a hight of 100 mV and a width of 1 ms. We measure everything here with respect, to the reversal potential. So the cell voltage relaxes back to zero (rather then approximately − 65 mV). All potentials are therefore measured with respect to this shifted zero.

Table 1 Parameters used in the model