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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.


Stationary activity of correlated networks.(A) Population firing rates and (B) distribution of single neuron firing rates for a network with P(k) ∼ k−2, k = [10, …, 500] for s = 1.2 from simulations (thin full and dotted lines) and theory (thick lines).
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pone.0121794.g003: Stationary activity of correlated networks.(A) Population firing rates and (B) distribution of single neuron firing rates for a network with P(k) ∼ k−2, k = [10, …, 500] for s = 1.2 from simulations (thin full and dotted lines) and theory (thick lines).

Mentions: Results for the population firing rates corresponding to a stimulus s = 1.2 are shown in Fig 3A together with the mean-field predictions. Theory and simulations agree very well. In the assortative network, firing rates of high-degree populations are raised and the ones of low-degree populations are lowered compared to the uncorrelated network. Disassortative networks show the opposite effect. The distribution of single-neuron firing rates P(ν) can be estimated directly from the population means, if one assumes that the firing rates of all neurons in a given k-population are equal. Then, each firing rate occurs NPin(k) times and this distribution of firing rates can simply be binned and normalized. This estimation is close to the actual firing rate distribution in the network (Fig 3B). Assortativity broadens the distribution, whereas disassortativity narrows it. Note, that the firing rate distributions appear to be power-law tailed in disassortative and uncorrelated networks only. Let us now consider the mean firing rate as a response to a sub-threshold stimulus s < 1 for different levels of assortativity. This is plotted in Fig 4 (open symbols) together with theoretical predictions (red lines). In an uncorrelated network for small s and shows a sharp transition to sustained activity at s ≈ 0.8, whereas assortative networks are active even for small s. The qualitative explanation is as follows. The neurons with low input degree eventually stop firing when their total input current becomes low. In uncorrelated networks this leads to a cascading failure of spiking of stronger connected neurons. In assortative networks the failure of neurons with low in-degree only leads to failure of the low-degree subnetwork, whereas high-degree subnetworks sustain their recurrent activity (Fig 5). This behavior is reminiscent of findings for percolation in complex networks: At low densities of links, assortative networks remain robust under random failure [48]. It is important to mention that the network may exhibit bistability for very low firing rates, where the mean-field solutions exhibit an additional unstable branch below the stable one that is shown in our results. In practice, bistability leads to hysteresis, where network dynamics depends on previous activity. This effect is discussed extensively in [16, 49]. However, we assume the network to operate in the stable upper branch exclusively by adjusting its sustained activity to changes in the stimulus instead of switching on and off. Networks with strong recurrent activity and the assortative networks considered below will sustain their activity even when the stimulus drops to zero.


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

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

Stationary activity of correlated networks.(A) Population firing rates and (B) distribution of single neuron firing rates for a network with P(k) ∼ k−2, k = [10, …, 500] for s = 1.2 from simulations (thin full and dotted lines) and theory (thick lines).
© Copyright Policy
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4482728&req=5

pone.0121794.g003: Stationary activity of correlated networks.(A) Population firing rates and (B) distribution of single neuron firing rates for a network with P(k) ∼ k−2, k = [10, …, 500] for s = 1.2 from simulations (thin full and dotted lines) and theory (thick lines).
Mentions: Results for the population firing rates corresponding to a stimulus s = 1.2 are shown in Fig 3A together with the mean-field predictions. Theory and simulations agree very well. In the assortative network, firing rates of high-degree populations are raised and the ones of low-degree populations are lowered compared to the uncorrelated network. Disassortative networks show the opposite effect. The distribution of single-neuron firing rates P(ν) can be estimated directly from the population means, if one assumes that the firing rates of all neurons in a given k-population are equal. Then, each firing rate occurs NPin(k) times and this distribution of firing rates can simply be binned and normalized. This estimation is close to the actual firing rate distribution in the network (Fig 3B). Assortativity broadens the distribution, whereas disassortativity narrows it. Note, that the firing rate distributions appear to be power-law tailed in disassortative and uncorrelated networks only. Let us now consider the mean firing rate as a response to a sub-threshold stimulus s < 1 for different levels of assortativity. This is plotted in Fig 4 (open symbols) together with theoretical predictions (red lines). In an uncorrelated network for small s and shows a sharp transition to sustained activity at s ≈ 0.8, whereas assortative networks are active even for small s. The qualitative explanation is as follows. The neurons with low input degree eventually stop firing when their total input current becomes low. In uncorrelated networks this leads to a cascading failure of spiking of stronger connected neurons. In assortative networks the failure of neurons with low in-degree only leads to failure of the low-degree subnetwork, whereas high-degree subnetworks sustain their recurrent activity (Fig 5). This behavior is reminiscent of findings for percolation in complex networks: At low densities of links, assortative networks remain robust under random failure [48]. It is important to mention that the network may exhibit bistability for very low firing rates, where the mean-field solutions exhibit an additional unstable branch below the stable one that is shown in our results. In practice, bistability leads to hysteresis, where network dynamics depends on previous activity. This effect is discussed extensively in [16, 49]. However, we assume the network to operate in the stable upper branch exclusively by adjusting its sustained activity to changes in the stimulus instead of switching on and off. Networks with strong recurrent activity and the assortative networks considered below will sustain their activity even when the stimulus drops to zero.

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.