Restoring the encoding properties of a stochastic neuron model by an exogenous noise

Abstract

Here we evaluate the possibility of improving the encoding properties of an impaired neuronal system by superimposing an exogenous noise to an external electric stimulation signal. The approach is based on the use of mathematical neuron models consisting of stochastic HH-like circuit, where the impairment of the endogenous presynaptic inputs is described as a subthreshold injected current and the exogenous stimulation signal is a sinusoidal voltage perturbation across the membrane. Our results indicate that a correlated Gaussian noise, added to the sinusoidal signal can significantly increase the encoding properties of the impaired system, through the Stochastic Resonance (SR) phenomenon. These results suggest that an exogenous noise, suitably tailored, could improve the efficacy of those stimulation techniques used in neuronal systems, where the presynaptic sensory neurons are impaired and have to be artificially bypassed.

Keywords: HH model; electric stimulation; exogenous noise; signal detection; single neuron; stochastic resonance.

Figure 1

Circuital representation of the neuron model. C_m is the specific membrane capacitance; g_L is the leakage specific conductance; g_Na and g_k are Sodium and Potassium specific conductances, voltage (V_m) dependent; I₀ is the specific stimulation current; V_Na, V_K, V_L are the reversal potentials for Sodium, Potassium, and leakage currents, respectively; V_ES and V_noise are the voltage perturbations, due to the exogenous electric signal and the exogenous Gaussian noise, superimposed to the physiological membrane voltage V; V_m is the total voltage between the intracellular and the extracellular space.

Figure 2

(A) Histogram showing the zero-mean Gaussian distribution of the exogenous voltage noise with power D = 2 mV²; the standard deviation (σ) of the distribution is equal to the square root of D. (B) Normalized Power Spectral Density of the exogenous voltage noise (blue line) compared with the theoretical Lorentzian spectrum with cutoff angular frequency ω_c = 2.5 ×10³ rad/s (red line).

Figure 3

Average PSD with the associated standard error (R = 100) of the output sequence U(t) for the neuron model with patch area 300 μm² and I₀ = 4 μA/cm² (A) or I₀ = 7 μA/cm² (B). The applied exogenous signal is a sinusoid at 150 Hz, 500 μV of amplitude.

Figure 4

Mean number of spikes per second and standard error calculated over the frequency of the input sinusoidal signal vs. the input constant current I₀ for patch areas: 200 (blue line), 300 (purple line), and 600 (orange line) μm².

Figure 5

Mean SNR and standard error (R = 100) vs. the signal frequency for I₀ = 7 μA/cm² and membrane patches of 200, 300, and 600 μm²; the signal is a sinusoid with amplitude V_ES = 500 μV and frequency ranging from 10 to 500 Hz.

Figure 6

Mean SNR and standard error (R = 100) vs. the signal frequency for I₀ = 2 and 4 μA/cm² and membrane patches of 200 μm² (A), 300 μm² (B), and 600 μm² (C); the signal is a sinusoid with amplitude V_ES = 500 μV and frequency ranging from 10 to 500 Hz.

Figure 7

Mean SNR and standard error (R = 300) as a function of the variance of the exogenous noise (D) for I₀ = 2 μA/cm² and membrane patches of 200 (A) and 300 μm² (B), and for I₀ = 4 μA/cm² and membrane patches of 300 μm² (C) and 600 μm² (D); the signal is a sinusoid with amplitude V_ES = 500 μV and frequency f = 150 Hz; the exogenous noise is a zero mean Gaussian process with a Lorentzian spectrum (ω_c = 2.5×10³ rad/s).

Figure 8

Mean number of spikes per second and standard error (R = 300) as a function of the variance of the exogenous noise (D) for I₀ = 2 μA/cm² and membrane patches of 200 μm² (solid blue line) and 300 μm² (solid purple line), and for I₀ = 4 μA/cm² and membrane patches of 300 μm² (dashed purple line) and 600 μm² (dashed orange line); the signal is a sinusoid with amplitude V_ES = 500 μV and frequency f = 150 Hz; the exogenous noise is a zero mean Gaussian process with a Lorentzian spectrum (ω_c = 2.5×10³ rad/s).