of Weak Signal Detection in Parallel Arrays of Integrate-and-Fire Neurons by Negative Spatial Correlation

(1)

for 1 ≤ i ≤ N, where v _idenotes the membrane voltage of neuron i, μ is a dc component in the noisy synaptic input and τ is the membrane time constant for the subthreshold dynamics. We adopt the standard threshold-spike-reset condition [8]: a spike is emitted whenever v _i(t) = V _T, and after that the voltage is reset and held at v _i(t ⁺) = V _Rduring a refractory period τ _R. Additionally, s(t) is a common component in the input and the Gaussian white noise η _i(t) (1 ≤ i, j ≤ N) models the internal stochastic component for the ith neuron. For each neuron, the output spike train y _i(t) = ∑ _ks(t − t _i^k) with t _i^kbeing the kth spike time is of interest, and the average spike train $y(t)=\frac{1}{N}{\displaystyle {\sum}_i{y}_i(t)}$ is taken as the output for summing parallel array. We assume that for 1 ≤ i, j ≤ N there is

$\left\langle {\eta}_i(t){\eta}_j\left(t+\tau \right)\right\rangle =\left[{\delta}_{i,j}+c{\delta}_{i,i+1}\right]\delta \left(\tau \right)$

(2)

with c being a tunable nearest-neighborhood correlation coefficient.

Since the local spatial correlation does not affect the response of each neuron, it is necessary to obtain the spectral statistics ${G}_{\it yy}\left(\omega \right)=\left\langle \tilde{y}\left(\omega \right){\tilde{y}}^{*}\left(\omega \right)\right\rangle$ for the ensemble output spike train, where $\tilde{y}\left(\omega \right)=\frac{1}{\sqrt{T}}{\displaystyle \underset{0}{\overset{T}{\int }} {\it dte}^{i\omega t}\left(y(t)-{r}_0(D)\right)}$ is the Fourier transform of the zero average output spike train with r ₀(D) being the stationary firing rate at the noise level D. For this aim, let us resort to technique of linear approximation [9–11]. According to linear approximation, each neuron can be regarded as a linear filter, and thus the frequency domain linear response for each neuron can be approximated as

${\tilde{y}}_i\left(\omega \right)={\tilde{y}}_{i,0}\left(\omega, D\right)+A\left(\omega, D\right)s\left(\omega \right)$

(3)

where ${\tilde{y}}_0\left(\omega \right)$ is the unperturbed part of stationary spectral density $\left\langle {\tilde{y}}_i^0\left(\omega \right){\tilde{y}}_i^{0*}\left(\omega \right)\right\rangle$ and A(ω, D) is linear susceptibility on the noise level of D. The stationary spectral density [12] and the linear susceptibility [13, 14] are explicitly given as

$A\left(\omega, \mu, D\right)=\frac{r(D) i\omega}{\sqrt{D}\left( i\omega -1\right)}\frac{{\tilde{D}}_{i\omega -1}\left(\frac{\mu -{v}_T}{\sqrt{D}}\right)-{e}^{\gamma }{\tilde{D}}_{i\omega -1}\left(\frac{\mu -{v}_R}{\sqrt{D}}\right)}{{\tilde{D}}_{i\omega}\left(\frac{\mu -{v}_T}{\sqrt{D}}\right)-{e}^{\gamma }{e}^{i\omega {\tau}_R}{\tilde{D}}_{i\omega}\left(\frac{\mu -{v}_R}{\sqrt{D}}\right)}$

(4)

${S}_0\left(\omega, \mu, D\right)\overset{\Delta}{=}\left\langle {\tilde{y}}_i^0\left(\omega \right){\tilde{y}}_i^{0*}\left(\omega \right)\right\rangle =r(D)\frac{{\left|{\tilde{D}}_{i\omega}\left(\frac{\mu -{v}_T}{\sqrt{D}}\right)\right|}^2-{e}^{2\gamma }{\left|{\tilde{D}}_{i\omega}\left(\frac{\mu -{v}_R}{\sqrt{D}}\right)\right|}^2}{{\left|{\tilde{D}}_{i\omega}\left(\frac{\mu -{v}_T}{\sqrt{D}}\right)-{e}^{\gamma }{e}^{i\omega {t}_R}{\tilde{D}}_{i\omega}\left(\frac{\mu -{v}_R}{\sqrt{D}}\right)\right|}^2}$

(5)

with $\gamma =\left[{v}_R^2-{v}_T^2+2\overline{\mu}\left({v}_T-{v}_R\right)\right]/4D$ . And then from Eq. (3), the auto-spectral density ${S}_{ii}\left(\omega \right)=\left\langle {\tilde{y}}_i\left(\omega \right){\tilde{y}}_i^{*}\left(\omega \right)\right\rangle$ for the ith neuron is obtained as

${S}_{i,i}\left(\omega \right)=\left|A\left(\omega, D\right)\right|{}^2{G}_{ss}\left(\omega \right)$