^{1}

^{2}

^{1}

^{2}

Edited by: Ad Aertsen, Albert Ludwigs University, Germany

Reviewed by: Olaf Sporns, Indiana University, USA; Tom Tetzlaff, Norwegian University of Life Sciences, Norway

*Correspondence: Alex Roxin, Theoretical Neurobiology of Cortical Circuits, Institut d'Investigacions Biomèdicas August Pi i Sunyer, Carrer Mallorca 183, Barcelona 08036, Spain. e-mail:

This is an open-access article subject to an exclusive license agreement between the authors and Frontiers Media SA, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are credited.

Neuronal network models often assume a fixed probability of connection between neurons. This assumption leads to random networks with binomial in-degree and out-degree distributions which are relatively narrow. Here I study the effect of broad degree distributions on network dynamics by interpolating between a binomial and a truncated power-law distribution for the in-degree and out-degree independently. This is done both for an inhibitory network (I network) as well as for the recurrent excitatory connections in a network of excitatory and inhibitory neurons (EI network). In both cases increasing the width of the in-degree distribution affects the global state of the network by driving transitions between asynchronous behavior and oscillations. This effect is reproduced in a simplified rate model which includes the heterogeneity in neuronal input due to the in-degree of cells. On the other hand, broadening the out-degree distribution is shown to increase the fraction of common inputs to pairs of neurons. This leads to increases in the amplitude of the cross-correlation (CC) of synaptic currents. In the case of the I network, despite strong oscillatory CCs in the currents, CCs of the membrane potential are low due to filtering and reset effects, leading to very weak CCs of the spike-count. In the asynchronous regime of the EI network, broadening the out-degree increases the amplitude of CCs in the recurrent excitatory currents, while CC of the total current is essentially unaffected as are pairwise spiking correlations. This is due to a dynamic balance between excitatory and inhibitory synaptic currents. In the oscillatory regime, changes in the out-degree can have a large effect on spiking correlations and even on the qualitative dynamical state of the network.

Network models of randomly connected spiking neurons have provided insight into the dynamics of real neuronal circuits. For example, networks operating in a balanced state in which large excitatory and inhibitory inputs cancel in the mean, can self-consistently and robustly account for the low, irregular discharge of neurons seen

The particular choice of random connectivity in these network models is one of convenience. The simplest random networks, known as Erdös–Rényi networks, can be generated with a single parameter

However, there is reason to go beyond Erdös–Rényi networks, which I will call standard random networks, and explore other types of random connectivity. Recent multiple intracellular recordings of neurons

In this paper I study the effect of in-degree and out-degree distributions on the spontaneous activity in networks of spiking neurons. Two distinct networks of randomly connected integrate-and-fire neurons are studied, the dynamics of both of which have been well characterized both numerically and analytically in the standard random case. The first network is purely inhibitory and exhibits fast oscillations with a period that is a few times the synaptic delay (Brunel and Hakim,

In neuronal networks, the probability of choosing a neuron in a network at random and finding it has _{in}_{out}_{in}_{out}

I generate networks of _{in}, _{out} and variances _{j}_{j}_{j}_{j}_{in} (_{out}) and variance

This measure goes to zero as _{i}_{i}_{i}_{i}_{i}_{i}

The in-degree and out-degree for any neuron

where ^{B}^{P}_{in}_{out}

Figure _{out}_{in}_{in}_{in}

_{in}_{in}_{0} and _{in}

For _{in}_{out}_{in}

with the reset condition _{i}^{+}) = _{reset}_{i}^{−}) ≥ _{rp} = 2 ms. Postsynaptic currents (PSCs) are modeled as delta functions _{ij}_{ij}_{E}_{I}_{ext}. PSCs are instantaneous with amplitude _{ext}_{reset}

Inhibitory network: _{I}_{ext}_{ext}

Excitatory–inhibitory network: _{E}_{I}_{ext}_{ext} = 8100 Hz.

In several figures CC of synaptic inputs and of spikes are shown. The measures I used to generate these figures are given here.

The spike train of a neuron _{i}

where the brackets denote a time average and the normalization is chosen so that the AC at zero-lag is equal to one.

In the network simulations, inputs consist of instantaneous jumps in the voltage of amplitude _{E}_{I}_{E,i}_{I,i}

where the brackets indicate a time average. The CC averaged over pairs is then

The spike train _{i}_{i}

I performed simulations of large networks of sparsely connected spiking neurons with different in-degree and out-degree distributions. Randomly connected networks were generated with parameters _{α}, _{α} = 0) and Power-law degree distributions (_{α} = 1) independently for the incoming and outgoing connections. For _{in}_{out}

The network consisted of 10,000 neurons driven by external, excitatory Poisson inputs and connected by inhibitory synapses modeled as a fixed delay followed by a jump in the postsynaptic voltage, see Section _{in}_{out}

The fast oscillations in the network activity were suppressed by broadening the in-degree (increasing _{in}_{out}_{in}_{out}

Figure _{in}_{out}_{in}_{out}

_{in}_{out}

The effect of the in-degree can be captured in an extension of a rate model invoked to capture the generation of fast oscillations in inhibitory networks (Roxin et al.,

The equation is

where _{in}

The steady state meanfield solution is given by 〈

The linear stability of the steady state solution depends only on the meanfield 〈

where

For simplicity I first consider the case of a threshold linear transfer function, Φ(_{+}, i.e., Φ(

The mean activity increases as

The effective gain function is

It can be seen upon inspection of Eq.

Figure _{in}_{in}

_{in}_{out}

The rate model Eq.

To quantify the above intuitive argument, if

where _{0} is the steady state solution for _{2}/_{3} ∼

Pairwise spiking correlations in neuronal networks can arise from various sources including direct synaptic connections between neurons as well as shared input (Shadlen and Newsome, _{out} is varied since the mean number of connections _{out} is fixed. However, the number of shared inputs is strongly influenced not only by the mean out-degree, but also by its variance _{l}

In the simulations conducted here, increasing _{out}_{f}

Here, despite large CCs in the currents, the pairwise CC of the membrane potential is very weak. This is shown in Figure

_{in}_{out}_{in}_{out}

Figure _{in}_{out}

Finally, Figure _{in}_{out}_{out}_{in}

The network consisted of 10,000 excitatory neurons and 2500 inhibitory neurons driven by external, excitatory Poisson inputs and connected by synapses modeled as a fixed delay followed by a jump in the postsynaptic voltage, see Section _{in}_{out}_{in}_{out}

Slow (25 Hz) oscillations emerged as the in-degree was broadened, see Figure

Figure

_{in}_{out}

As before, one can understand how the in-degree affects oscillations in the network by studying a rate model. The model now includes two coupled equations

where _{e}_{i}

I assume that the external input to the inhibitory population _{i}

where

where

Again I look at the simple case of a threshold linear transfer function. Choosing

the stability of which is determined by

By inspection, it is clear that _{ee}^{α} gives

where

_{ee}

Figure _{ee}_{in}_{in}

_{ee}_{ii}_{e}

In this network one would expect increasing the variance of the out-degree distribution to lead to an increase in the amplitude of CC in the recurrent excitatory input. This is indeed the case, as can be seen in Figure _{in}_{out}_{out}

_{out}_{in}

Figure _{in}_{out}_{in}

In fact, broadening the out-degree distribution can even lead to qualitative changes in the dynamical state of the network, as long as the system is poised near a bifurcation. This is shown in Figure _{in}_{out}_{out}

_{in}_{out}_{out}

I have conducted numerical simulations of two canonical networks as a function of the in-degree and out-degree distributions of the network connectivity. For both the purely inhibitory (I), as well as the EI networks, it was the in-degree which most strongly affected the global, dynamical state of the network. In both cases, increasing the variance of the in-degree drove a transition in the dynamical state: in the I network oscillations were abolished while in the EI network, oscillations were generated when the E-to-E in-degree was broadened. The analysis of a simple rate model, suggests that these transitions can be understood as the effect of in-degree on the effective input–output gain of the network. Specifically, in a standard random network with identical neurons, the gain of the network in the spontaneous state can be expressed as the slope of the non-linear transfer function which converts the total input to neurons into an output, e.g., a firing rate. This is the approximation made in a standard, scalar rate equation. A high gain makes the network more susceptible to instabilities, e.g., oscillations. In the case of a network with a broad in-degree distribution, each neuron receives a different level of input, and the effective gain is now the gain of each neuron, averaged over neurons and weighted by the in-degree. In this way the stability of the spontaneous state may depend crucially on the shape of the transfer function. The transfer function for integrate-and-fire neurons in the fluctuation-driven regime is concave-up. For this type of transfer function, the simple rate equation predicts that oscillations will be suppressed in the I network and enhanced in the EI network, in agreement with the network simulations. It has been shown that the single-cell fI curve of cortical neurons operating in the fluctuation-driven regime is well approximated by a power-law with power greater than one (Hansel and van Vreeswijk,

The out-degree distribution determines the amount of common, recurrent input to pairs of neurons, and as such may be expected to affect pairwise spiking correlations. Yet predicting spike correlations based on knowledge of input correlations has proven a non-trivial task and can depend crucially on firing rate, external noise amplitude, and the global dynamical state of the network to name a few factors (de la Rocha et al.,

The in-degree and out-degree distributions alone may not be sufficient to characterize the connectivity in real neuronal networks. As an example, while broad degree distributions lead to an over-representation of triplet motifs compared to the standard random network, e.g., see Figure

How should one proceed in investigating the role of connectivity on network dynamics? As mentioned in the previous paragraph, there are other statistical measures of network connectivity which allow one to characterize network topology and conduct parametric analyses, e.g., motifs. No one measure is more principled than another and they are not, in general, independent. Parametric studies, such as this one, can shed light on the role of certain statistical features of network topology in shaping dynamics. Specifically, for networks of sparsely coupled spiking neurons, the width of the in-degree strongly affects the global dynamical state, while the width of the out-degree affects pairwise correlations in the synaptic currents. An alternative and more ambitious approach would allow synaptic connections to evolve according to appropriate plasticity rules. In this way, the network topology would be shaped by stimulus-driven inputs and could then be related to the functionality of the network itself. Additional constraints, such as minimizing wiring length for fixed functionality, may lead to topologies which more closely resemble those measured in the brain (Chklovskii,

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This section provides details to the analysis of the inhibitory rate model described in the main text.

Here I take _{+}. The meanfield steady state solution of the rate equation, Eq. _{cr}_{cr}^{*} and so ^{*})〈

where

The linear stability of the solution 〈

where setting

The steady state meanfield solution is expanded as 〈_{0} + _{1} + ^{2}_{2} and the dependence on in-degree is kept general, i.e., _{1} since linear changes in ^{2},

The stability function

Taking Φ(^{α} gives Eq.

This section provides details to the analysis of the excitatory–inhibitory rate model described in the main text.

Here I take _{+}. The meanfield steady state solution of the rate equation, Eq. _{cr}^{*} and so ^{*} is found from the condition −^{*})〈

Setting

The linear stability of the solution 〈_{e}_{e}_{i}_{i}_{i})^{i}^{wt}

where setting

The steady state meanfield solution is expanded as 〈_{e}_{0} + _{1} + ^{2}_{2} and the dependence on in-degree is kept general, i.e., _{1} since linear changes in ^{2} gives Eq.

A more detailed and general description of the role of filtering on CCs can be found in Tetzlaff et al. (

where the mean input has already been subtracted off, _{0} are the oscillation amplitude and frequency respectively, and _{i}_{j}

where the brackets indicate an average over time. The voltage obeys the following stochastic differential equation

where _{i} is a Gaussian random variable with mean zero and unit variance with _{i}_{j}_{ij}

where η_{i}_{i}_{j}_{ij}e^{−(t − t′)/τ}^{2} where _{i}_{j}

Taking the ratio of the voltage CC to the current CC gives

which is less than one for _{0} = 0.14/ms gives (

I thank Duane Nykamp and Jaime de la Rocha for very useful discussions. I thank Rita Almeida, Jaime de la Rocha, Anders Ledberg, Duane Nykamp, and Klaus Wimmer for a careful reading of the manuscript. I would like to thank the Gatsby Foundation, the Kavli Foundation and the Sloan–Swartz Foundation for their support. Alex Roxin was a recipient of a fellowship from IDIBAPS Post-Doctoral Programme that was cofunded by the EC under the BIOTRACK project (contract number PCOFUND-GA-2008-229673). This work was funded by the Spanish Ministry of Science and Innovation (ref. BFU2009-09537), and the European Regional Development Fund.