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Degree Correlations Optimize Neuronal Network Sensitivity to Sub-Threshold Stimuli.

Schmeltzer C, Kihara AH, Sokolov IM, Rüdiger S - PLoS ONE (2015)

Bottom Line: We devise a method to calculate firing rates from a self-consistent system of equations taking into account the degree distribution and degree correlations in the network.We show that assortative degree correlations strongly improve the sensitivity for weak stimuli and propose that such networks possess an advantage in signal processing.We moreover find that there exists an optimum in assortativity at an intermediate level leading to a maximum in input/output mutual information.

View Article: PubMed Central - PubMed

Affiliation: Institut für Physik, Humboldt-Universität zu Berlin, Germany.

ABSTRACT
Information processing in the brain crucially depends on the topology of the neuronal connections. We investigate how the topology influences the response of a population of leaky integrate-and-fire neurons to a stimulus. We devise a method to calculate firing rates from a self-consistent system of equations taking into account the degree distribution and degree correlations in the network. We show that assortative degree correlations strongly improve the sensitivity for weak stimuli and propose that such networks possess an advantage in signal processing. We moreover find that there exists an optimum in assortativity at an intermediate level leading to a maximum in input/output mutual information.

No MeSH data available.


Response curves of three model networks with power-law in-degree distribution Pin(k) ∼ kα, k = [10, … 500] and slightly different exponents α.For each network we show the response curves for four different levels of assortativity (thin black lines). The thick light red lines indicate ±1SD of the noise of the output from n = 5000 neurons. (A) Model network with large negative exponent of α = −2.3 and small mean degree. The network responses to sub-threshold stimuli are very weak due to small recurrent activity. The uncorrelated network (p = 0) begins to fire above the stimulus threshold of s > 0.8. This threshold is reduced for increasing assortativity, and the response becomes very noisy. (B) Model network with intermediate exponent α = −2 and intermediate mean-degree. The network responses are stronger than for α = −2.3, but the stimulus threshold for the uncorrelated network is similar at s ≃ 0.8. (C) Model network with small negative exponent and large mean degree. The network fires at high rates and has a low stimulus threshold s ≃ 0.55 due to strong recurrent activity.
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pone.0121794.g008: Response curves of three model networks with power-law in-degree distribution Pin(k) ∼ kα, k = [10, … 500] and slightly different exponents α.For each network we show the response curves for four different levels of assortativity (thin black lines). The thick light red lines indicate ±1SD of the noise of the output from n = 5000 neurons. (A) Model network with large negative exponent of α = −2.3 and small mean degree. The network responses to sub-threshold stimuli are very weak due to small recurrent activity. The uncorrelated network (p = 0) begins to fire above the stimulus threshold of s > 0.8. This threshold is reduced for increasing assortativity, and the response becomes very noisy. (B) Model network with intermediate exponent α = −2 and intermediate mean-degree. The network responses are stronger than for α = −2.3, but the stimulus threshold for the uncorrelated network is similar at s ≃ 0.8. (C) Model network with small negative exponent and large mean degree. The network fires at high rates and has a low stimulus threshold s ≃ 0.55 due to strong recurrent activity.

Mentions: In Fig 8 we show the response curves and the associated standard deviation of the output for n = 5000 of three model networks with power-law distribution between kmin = 10 and kmax = 500 and increasing exponents α = (−2.3, −2, −1.7). The response curves and variances of the firing rate distributions were calculated with the mean-field approach. The joint degree distributions Nkk′ were sampled from the adjacency matrices of the networks. The means ⟨k⟩ and variances of the degree distributions are shown in Table 1. The first network with large negative exponent α = −2.3 and small mean degree fires at very low rates (Fig 8A), which corresponds to a low response variability. Assortativity changes the shape of the response curve and decreases the stimulus threshold, but strongly increases the output noise. The network with small negative exponent α = −1.7 and large mean degree fires at high rates, and the stimulus threshold for the uncorrelated network is shifted to a lower value, s ≃ 0.55 (Fig 8C). Hence, this network has a large response variability even when no in-degree correlations are present. For this network, weaker levels of assortativity are sufficient to reshape the response curve and lower the stimulus threshold even further. For all three networks, the noise level is strongly increased by assortativity.


Degree Correlations Optimize Neuronal Network Sensitivity to Sub-Threshold Stimuli.

Schmeltzer C, Kihara AH, Sokolov IM, Rüdiger S - PLoS ONE (2015)

Response curves of three model networks with power-law in-degree distribution Pin(k) ∼ kα, k = [10, … 500] and slightly different exponents α.For each network we show the response curves for four different levels of assortativity (thin black lines). The thick light red lines indicate ±1SD of the noise of the output from n = 5000 neurons. (A) Model network with large negative exponent of α = −2.3 and small mean degree. The network responses to sub-threshold stimuli are very weak due to small recurrent activity. The uncorrelated network (p = 0) begins to fire above the stimulus threshold of s > 0.8. This threshold is reduced for increasing assortativity, and the response becomes very noisy. (B) Model network with intermediate exponent α = −2 and intermediate mean-degree. The network responses are stronger than for α = −2.3, but the stimulus threshold for the uncorrelated network is similar at s ≃ 0.8. (C) Model network with small negative exponent and large mean degree. The network fires at high rates and has a low stimulus threshold s ≃ 0.55 due to strong recurrent activity.
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pone.0121794.g008: Response curves of three model networks with power-law in-degree distribution Pin(k) ∼ kα, k = [10, … 500] and slightly different exponents α.For each network we show the response curves for four different levels of assortativity (thin black lines). The thick light red lines indicate ±1SD of the noise of the output from n = 5000 neurons. (A) Model network with large negative exponent of α = −2.3 and small mean degree. The network responses to sub-threshold stimuli are very weak due to small recurrent activity. The uncorrelated network (p = 0) begins to fire above the stimulus threshold of s > 0.8. This threshold is reduced for increasing assortativity, and the response becomes very noisy. (B) Model network with intermediate exponent α = −2 and intermediate mean-degree. The network responses are stronger than for α = −2.3, but the stimulus threshold for the uncorrelated network is similar at s ≃ 0.8. (C) Model network with small negative exponent and large mean degree. The network fires at high rates and has a low stimulus threshold s ≃ 0.55 due to strong recurrent activity.
Mentions: In Fig 8 we show the response curves and the associated standard deviation of the output for n = 5000 of three model networks with power-law distribution between kmin = 10 and kmax = 500 and increasing exponents α = (−2.3, −2, −1.7). The response curves and variances of the firing rate distributions were calculated with the mean-field approach. The joint degree distributions Nkk′ were sampled from the adjacency matrices of the networks. The means ⟨k⟩ and variances of the degree distributions are shown in Table 1. The first network with large negative exponent α = −2.3 and small mean degree fires at very low rates (Fig 8A), which corresponds to a low response variability. Assortativity changes the shape of the response curve and decreases the stimulus threshold, but strongly increases the output noise. The network with small negative exponent α = −1.7 and large mean degree fires at high rates, and the stimulus threshold for the uncorrelated network is shifted to a lower value, s ≃ 0.55 (Fig 8C). Hence, this network has a large response variability even when no in-degree correlations are present. For this network, weaker levels of assortativity are sufficient to reshape the response curve and lower the stimulus threshold even further. For all three networks, the noise level is strongly increased by assortativity.

Bottom Line: We devise a method to calculate firing rates from a self-consistent system of equations taking into account the degree distribution and degree correlations in the network.We show that assortative degree correlations strongly improve the sensitivity for weak stimuli and propose that such networks possess an advantage in signal processing.We moreover find that there exists an optimum in assortativity at an intermediate level leading to a maximum in input/output mutual information.

View Article: PubMed Central - PubMed

Affiliation: Institut für Physik, Humboldt-Universität zu Berlin, Germany.

ABSTRACT
Information processing in the brain crucially depends on the topology of the neuronal connections. We investigate how the topology influences the response of a population of leaky integrate-and-fire neurons to a stimulus. We devise a method to calculate firing rates from a self-consistent system of equations taking into account the degree distribution and degree correlations in the network. We show that assortative degree correlations strongly improve the sensitivity for weak stimuli and propose that such networks possess an advantage in signal processing. We moreover find that there exists an optimum in assortativity at an intermediate level leading to a maximum in input/output mutual information.

No MeSH data available.