Noise-enhanced nonlinear response and the role of modular structure for signal detection in neuronal networks

We show that sensory noise can enhance the nonlinear response of neuronal networks, and when delivered together with a weak signal, it improves the signal detection by the network. We reveal this phenomenon in neuronal networks that are in a dynamical state preceding a saddle-node bifurcation corresponding to the appearance of sustained network oscillations. In this state, even a weak subthreshold pulse can evoke a large-amplitude oscillation of neuronal activity. The signal-to-noise ratio reaches a maximum at an optimum level of sensory noise, manifesting stochastic resonance (SR) at the population level. We demonstrate SR by use of simulations and numerical integration of rate equations in a cortical model. Using this model, we mimic the experiments of Gluckman et al. [Phys. Rev. Lett. 77, 4098 (1996)] that have given evidence of SR in mammalian brain. We also study neuronal networks in which neurons are grouped in modules and every module works in the regime of SR. We find that even a few modules can strongly enhance the reliability of signal detection in comparison with the case when a modular organization is absent.

Figure 1

Phase diagram of the cortical model in dependence on the flow intensity $\langle n\rangle$ of random spikes bombarding neurons in the case when excitatory neurons respond faster to input than inhibitory neurons ($\alpha =0.7$) (adapted from [30]). Stochastic resonance takes place in the region $n_{c1}<\langle n\rangle <n_{c2}$ preceding the saddle-node bifurcation at $\langle n\rangle =n_{c2}$. Above $n_{c2}$ sustained network oscillations appear.

Figure 2

In the absence of sensory noise, a periodic sensory signal ($S$) with the amplitude $A_s=4.5\times {10}^{-3}$ [see panel (a)] generates a weak perturbation of the excitatory population activity ${\rho }_e$ that can hardly be identified in panel (b). However, the addition of sensory noise $\xi$ with the mean amplitude $A_{\xi }=4.3\times {10}^{-2}$ [see panel (c)], which is about 10 times larger than the signal's amplitude $A_s$, results in neuronal activity with single sharp oscillations shown in panel (d). The single sharp oscillations appear preferentially near the peaks of the sensory signal. Network parameters: $c=1000$, $\Omega =30$, $g_i=0.25$, $J_i=-3J_e$, ${\sigma }^2=10$, $\langle n\rangle =10$, $\alpha =0.7$, $g_s=0.1$, and $f_s=1.25$ Hz. Time $t$ is in units $1/{\mu }_e$.

Figure 3

Burst probability density (BPD) versus the phase $\varphi$ of the sinusoidal sensory signal from the numerical integration of Eq. (8). (a) The BPD versus $\varphi$ in the presence of sensory noise with the mean amplitude $A_{\xi }=4.3\times {10}^{-2}$ when the signal is very weak ($A_s\ll A_{\xi }$). (b) BPD versus $\varphi$ at sensory noise $A_{\xi }=2.4\times {10}^{-2}$. (c) BPD at the optimal level of sensory noise $A_{\xi }=4.3\times {10}^{-2}$. (d) BPD at strong sensory noise, $A_{\xi }=7.0\times {10}^{-2}$. The signal's amplitude $A_s=4.5\times {10}^{-3}$ is the same for (b), (c), and (d). The data were obtained by averaging over 2500 periods of the signal. The dashed lines represent the signal versus $\varphi$. Other parameters are the same as in Fig. 2.

Figure 4

Signal-to-noise ratio (SNR) versus the mean amplitude $A_{\xi }$ of the sensory noise in the cortical model from numerical integration of Eq. (8). SNR is in decibel [$10{log}_{10}\left(\text{SNR}\right)$]. Error bars were estimated from rms distribution of 10 measurements. The bar length is equal to twice the standard deviation. The middle point of the bar corresponds to the mean value of SNR. Parameters are the same as those in Fig. 2.

Figure 5

Response of the cortical model of neuronal networks to the sinusoidal sensory signal $S\left(t\right)$ at different levels $\langle \xi \rangle$ of sensory noise: (a) $\langle \xi \rangle =5.0$; (b) $\langle \xi \rangle =5.5$; (c) $\langle \xi \rangle =6.0$; (d) $\langle \xi \rangle =7.0$; (e) sinusoidal signal $S\left(t\right)$; (f) $\langle \xi \rangle =7.5$. Parameters: $A_s=4.5$ and ${\sigma }_{sn}^2=5$. Other parameters in simulations are the same as in Fig. 2.

Figure 6

Power spectral density (PSD) of the cortical model in which a small fraction ($g_s=0.1$) of excitatory neurons is stimulated by a sinusoidal signal [Eq. (9)] with frequency $f_s=1.25$ Hz in the presence of sensory noise with $\langle \xi \rangle =7.0$. (a) PSD versus the frequency $f$, linear scale; (b) PSD versus $f$, log-log scale. (c) Signal-to-noise (SNR) ratio versus $\langle \xi \rangle$. SNR is in decibel [$10{log}_{10}\left(\text{SNR}\right)$]. Parameters in simulations are the same as in Figs. 2 and 5.

Figure 7

Signal detection in neuronal networks with modular structure from numerical integration of Eq. (3). (a) A signal together with noise is delivered to four modules 1, 2, 3, and 4. Responses of these modules are averaged in a module denoted as $\Sigma$. (b) S represents the signal “ola” sent to these four modules. This signal with noise generates output signals 1, 2, 3, and 4 from the respective modules. The signal $\Sigma$ represents the average over the output signals.

Figure 8

(a) Probability $p\left(n\right)$ that a signal's pulse is detected by a module of size $N/n$ versus $n$ ($N=50400$ in our simulations). (b) Probability $\Pi \left(n\right)$ of the signal detection in a network with $n$ modules. In panels (a) and (b), symbols represent results of two stimulation methods: (1) each neuron in every module receives independent sensory noise but the same pulsed signal (triangles); (2) neurons in the modules receive the same sensory noise and the same pulsed signal (squares). Parameters of the signal+noise input are in the text. Other parameters are the same as in Figs. 2 and 5.

Figure 9

(a) Train of random rectangular pulses as a sensory signal. Panels (b)–(e) show stimulated neuronal activity of one of $n$ modules of size $N_m=N/n$ for $N=50400$: (b) $n=2$ ($N_m=25200$); (c) $n=10$ ($N_m=5040$); (d) $n=30$ ($N_m=1680$); (e) $n=40$ ($N_m=1260$). Neurons receive independent sensory noise together with the signal. Other parameters are the same as in Fig. 8.