The role of synaptic facilitation in spike coincidence detection

Abstract

Using a realistic model of activity dependent dynamical synapse, which includes both depressing and facilitating mechanisms, we study the conditions in which a postsynaptic neuron efficiently detects temporal coincidences of spikes which arrive from N different presynaptic neurons at certain frequency f. A numerical and analytical treatment of that system shows that: (1) facilitation enhances the detection of correlated signals arriving from a subset of presynaptic excitatory neurons, and (2) the presence of facilitation yields to a better detection of firing rate changes in the presynaptic activity. We also observed that facilitation determines the existence of an optimal input frequency which allows the best performance for a wide (maximum) range of the neuron firing threshold. This optimal frequency can be controlled by means of facilitation parameters. Finally, we show that these results are robust even for very noisy signals and in the presence of synaptic fluctuations produced by the stochastic release of neurotransmitters.

Appendix

1.1 Analytical derivation of the error function E(f,V _th )

In this section we derived analytical expressions for the functions appearing in the definition of the error function (5) used to obtain theoretically the regions for good spike CD in the (f,V _th) parameter space.

First, we assume that the total presynaptic current can be divided in two terms: a signal term containing the correlated embedded signal and a noise term formed by the background of uncorrelated spikes.

1.1.1 Noise contribution

To take into account the noise generated by N − M uncorrelated spikes trains, we assume that the current at time t = t ^* + τ generated by a single spike arriving to the synapse i at time t ^* is given by

$$ I_{i}(\tau,t^*)=I_{peak}\exp(-\tau/\tau_{in}) $$

(20)

where I _peak represents the averaged stationary EPSC amplitude obtained after stimulation with a periodic spike train, assumption that we also suppose valid for a Poisson distributed spike train. After this consideration, one easily obtains from Eqs. (1–3) that

$$ I_{peak}=A_{SE}\frac{U_\infty(1-\exp(-1/f\tau_{rec}))}{1-(1-U_\infty)\exp(-1/f\tau_{rec})} \label{peak} $$

(21)

with U _∞ = u _∞(1 − U _SE ) + U _SE , where u _∞ is the value of u(t) in the stationary state (t→ ∞). For a periodic spike train, u _∞ is given by

$$ u_\infty=U_{SE}\frac{\exp(-1/f\tau_{fac})}{1-(1-U_{SE})\exp(-1/f\tau_{fac})}. $$

(22)

We can compute the mean noise contribution of the current and fluctuations using the standard expressions

$$\begin{array}{*{20}l} {{I_{{noise}} \equiv {\left\langle I \right\rangle },} \hfill} \\ {{\sigma ^{2}_{{I_{{noise}} }} \equiv {\left\langle {I^{2} } \right\rangle } - {\left\langle I \right\rangle }^{2} } \hfill} \\ \end{array} $$

(23)

From these definitions and using the central limit theorem we obtain

$$ I_{noise}\!=\!(N-M)A_{SE}f\tau_{in}U_{\infty}\frac{1-\exp(-1/f\tau_{rec})}{1\! -\! (1-U_\infty)\exp(\! -1/f\tau_{rec})} $$

(24)

where we assumed that τ _in  ≪ τ _rec .

If we neglect fluctuations ($\sigma_{I_{noise}}=0$), we can write V _noise  = R _in I _noise . Using this expression one can compute N _falses taking into account that false firing occurs when V _noise  > V _th so by a direct integration of Eq. (4) in a period of time T gives N _falses  ≈ T/{τ _ref  − τ _m ln (1 − V _th /V _noise )} (Koch 1999). Now using that f = N _inputs /T, we finally obtain as in Pantic et al. (2003)

$$ N_{falses}=\frac{\theta(V_{noise}-V_{th})N_{inputs}}{f(\tau_{ref}-\tau_{m}\ln(1-V_{th}/V_{noise}))} \label{falses} $$

(25)

where θ(x) is the Heaviside step function, which takes into account that for V _noise  < V _th N _falses  = 0.

To take into account fluctuations of I _noise one can use the so called hazard function approximation (Plesser and Gerstner 2000) but it has been reported that it gives the same results as those obtained using the formula (25) for high frequencies and, on the contrary to the expression (25), it does not work properly for small frequencies (Pantic et al. 2003). Therefore, hereafter we will neglect fluctuations in I _noise and use Eq. (25) as an approximatively valid expression to analytically compute N _falses .

1.1.2 Signal contribution

To analyse the signal contribution (arising from M coincident spikes) we used the same method developed in Pantic et al. (2003) for the case of only depressing synapses. That is, assuming that V(0;t ^*) is the membrane potential at t = t ^* when M coincident spikes arrive, by direct integration of the Eq. (4) the membrane potential at time t = t ^* + τ is

$$ V(\tau;t^{*})=e^{\tau/\tau_{m}}\left \{ V(0;t^*) + \frac{R_{in}M I_{peak}}{\tau_{m}\alpha}[e^{\alpha\tau}-1]\right \} \label{voltage} $$

(26)

where $\alpha= \frac{\tau_{in}-\tau_{m}}{\tau_{in}\tau_{m}}$ and I _peak is given by Eq. (21) including all the effects due to synaptic depression and facilitation. If the next signal event (M coincident spikes) occurs at t = t′ one can obtain the following recurrence relation:

$$ V(0;t')=e^{\Delta t/\tau_{m}}\left \{V(0;t^*) + \frac{R_{in}M I_{peak}}{\tau_{m}\alpha}[e^{\alpha\Delta t}-1]\right \} $$

(27)

with Δt = t′ − t ^*, which allows for computing the stationary value for the membrane potential at the exact time of the signal event arrival (see also Kistler and van Hemmen 1999), that is:

$$ V_{st}=e^{-\Delta t/\tau_{m}}\frac{R_{in}M I_{peak}}{\tau_{m}\alpha}\frac{e^{\alpha \Delta t} -1}{(1-e^{-\Delta t/\tau_{m}})}. $$

(28)

We define V _signal as the maximum of the membrane potential reached between the arrival of two consecutive signal events separated by a time Δt. This can be easily computed from Eq. (26) with V(0,t ^*) replaced by V _st :

$$ V_{signal}=\Biggl[\frac{\tau_{m}(1-\exp(-1/f\tau_{m}))}{\tau_{in}(1-\exp(-1/f\tau_{in}))}\Biggr]^{\frac{\tau_{m}}{\tau_{in}-\tau_{m}}}R_{in}MI_{peak} $$

(29)

where we consider $\tau=\Delta t\backsimeq 1/f.$

The expression of V _signal allows for an evaluation of the number of failures assuming that N _failures  = N _inputs  − N _hits . Then, one obtains by direct integration of Eq. (4) and using the same reasoning as for N _falses case that

$$N_{{failures}} = N_{{inputs}} {\left[ {1 - \frac{{\theta {\left( {V_{{noise}} + V_{{signal}} - V_{{th}} } \right)}}}{{f{\left[ {\tau _{{ref}} - \tau _{m} \ln {\left( {1 - {{\left( {V_{{th}} - V_{{signal}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{{th}} - V_{{signal}} } \right)}} {V_{{noise}} }}} \right. \kern-\nulldelimiterspace} {V_{{noise}} }} \right)}} \right]}}}} \right]}$$

(30)

where we have considered a hit event every time V _noise  + V _signal reach V _th . Note that from Eq. (30) if V _noise  + V _signal  < V _th we will have N _failures  = N _inputs .

Expression for N _falses , N _failures allows to theoretically compute the number of errors in the CD maps.