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Wrote the paper: RMM MT. Designed research: RMM MT. Performed research: RMM MT.

The authors have declared that no competing interests exist.

Despite the current debate about the computational role of experimentally observed precise spike patterns it is still theoretically unclear under which conditions and how they may emerge in neural circuits. Here, we study spiking neural networks with non-additive dendritic interactions that were recently uncovered in single-neuron experiments. We show that supra-additive dendritic interactions enable the persistent propagation of synchronous activity already in purely random networks without superimposed structures and explain the mechanism underlying it. This study adds a novel perspective on the dynamics of networks with nonlinear interactions in general and presents a new viable mechanism for the occurrence of patterns of precisely timed spikes in recurrent networks.

Most nerve cells in neural circuits communicate by sending and receiving short stereotyped electrical pulses called action potentials or spikes. Recent neurophysiological experiments found that under certain conditions the neuronal dendrites (branched projections of the neuron that transmit inputs from other neurons to the cell body (soma)) process input spikes in a nonlinear way: If the inputs arrive within a time window of a few milliseconds, the dendrite can actively generate a dendritic spike that propagates to the neuronal soma and leads to a nonlinearly amplified response. This response is temporally highly precise. Here we consider an analytically tractable model of spiking neural circuits and study the impact of such dendritic nonlinearities on network activity. We find that synchronous spiking activity may robustly propagate through the network, even if it exhibits purely random connectivity without additionally superimposed structures. Such propagation may contribute to the generation of spike patterns that are currently discussed to encode information about internal states and external stimuli in neural circuits.

Patterns of spikes that are precisely timed within the millisecond range have been investigated and observed in a series of neurophysiological studies

During the last two decades, a branch of theoretical research has focused on the question how such precise timing could emerge. One prominent, possible explanation for the occurrence of precisely coordinated spiking is the existence of excitatorily coupled feed-forward structures, ‘synfire-chains’, which are superimposed on a network of otherwise random connectivity, e.g. through strongly enhanced synaptic connectivity

Here we show that nonlinear dendritic interactions, recently uncovered in neurophysiological experiments, offer a viable mechanism to support stable propagation of synchrony through random cortical circuits without additionally superimposed structures: Excitatory synaptic stimuli may not only superimpose linearly or sublinearly

In the following, we study consequences of coupling nonlinearities that are due to fast dendritic spikes onto the collective dynamics of recurrent neural networks. We find that, in contrast to linearly coupled networks, propagating synchronous activity may persist already in networks of simple neurons that have purely random connectivity and exhibit no additional structures. We conclude that the characteristic features of dendritic nonlinearity, in particular the amplification of (only) synchronous input and the induction of temporally precise output, predestine them to support the generation and propagation of persistent, highly synchronous spiking activity.

We investigate networks of integrate-and-fire neurons in the limit of fast response to incoming spikes and with nonlinear interactions (see

To account for nonlinear enhancement and saturation of synchronous excitatory inputs, we modulate the linear sum of the amplitudes of excitatory post-synaptic potentials (EPSPs) that arise simultaneously from different synapses by a nonlinear function

The modulation function maps the somatic peak EPSP expected from linear summation of inputs to the actual peak EPSP strength. In networks with additively coupled neurons (a), the modulation function is the identity. In networks with nonlinear dendritic enhancement of inputs (b), the modulation function is sigmoidal as found in physiological experiments. Supra-additivity sets in when the expected (linearly added) input strength reaches a threshold

In both additively and non-additively coupled sparse random recurrent networks, asynchronous irregular spiking activity constitutes a dynamical state typical for a wide range of parameters

How does a sparse random network respond to induced synchronous activity, initiated, e.g., by external stimuli? We compared the responses in networks with purely linear, additive coupling to those where the excitatory inputs cooperate supra-additively. For linearly coupled networks we find that pulse sizes in chains of synchronous spiking activity quickly reduce to the level of spontaneous synchronization and the chains rapidly die out (cf.

The figure illustrates the temporal evolution of propagating synchrony as typical for large ranges of parameters in conventional networks (a,b,c) and in networks incorporating nonlinear dendritic interactions (d,e,f). Panels (c,f) show the spiking activity of the first

The parameter scans illustrate this by varying the mean total input strengths

Persistent propagation of synchrony is robust against parameter changes. We estimate a range of coupling strengths where persistent propagation of synchrony occurs in linearly and in nonlinearly coupled networks in

The mechanisms underlying this persistent propagation of synchrony can be intuitively understood. Sequences with small groups of synchronized neurons behave as for linear, additive coupling, i.e. they usually extinguish after a few steps, so there is no persistent spontaneous propagation and irregular background dynamics for the entire network is stable. If larger groups of neurons send spikes simultaneously, their postsynaptic neurons receive sufficiently many excitatory inputs so that the nonlinearities become effective. Since the inhibitory couplings add only linearly, excitatory input surpasses inhibitory input for a larger fraction of postsynaptic neurons than in a linearly coupled network. This causes more neurons to fire in response to the synchronous pulse; the number of neurons synchronized in each step of the chain grows. If synchronous pulses become too large, saturation becomes important and excitation becomes less efficient compared to inhibition. Further, many neurons are refractory. This implies that less neurons are excited in response to overly large groups of synchronously spiking neurons; consequently the group size is reduced. In addition, fluctuations in groups sizes occur due to the randomness of the network connectivity and the distribution of membrane potentials during pulse reception. These qualitative mechanisms keep the group sizes substantially large and fluctuating within a certain range.

To quantitatively understand the mechanisms underlying persistent propagation of synchrony and to determine the group sizes which initiate and take part in persistent propagation, we studied the evolution of propagating synchrony both analytically and numerically (see

Numerically derived probability distributions

For networks of linearly coupled neurons, each synchronous group with

In contrast, nonlinear supra-additive excitatory coupling enables persistent propagation of activity with a substantial number of neurons synchronized. The sizes of the propagating synchronous pulses are of the order of a typical size

The different dynamics for linearly and nonlinearly coupled networks can also be understood by approximating the stochastic dynamics by a deterministic iterative map derived from interpolating between the values of

Taken together, the theory for nonlinearly coupled networks predicts persistent propagation of synchronous activity in a typical range of pulse sizes and a decay that is possible only due to fluctuations. This agrees with the numerical observations (

In summary, we presented a theoretical analysis and numerical simulations of recurrent networks of spiking neurons with nonlinear dendritic interactions. The results indicate that networks with nonlinear dendritic interactions are capable of generating persistent propagation of synchronous spiking activity even if the network is purely randomly connected and has no additional structural features.

Theoretical studies on active dendrites mainly considered single neurons. Simulations of neuron models with detailed channel density and morphology showed dendritic spike generation in agreement with neurobiological experiments

The present study now shows that fast dendritic spikes can lead to

Our study uses a model that is appropriate for quantitative numerical analysis of larger networks and at the same time allows analytical predictions that yield further insights into the dynamics of recurrent networks. The theoretical predictions made are based on mean field arguments, strictly valid only in the limit of infinite network size

The current study contributes to a new field of research that focuses on neural networks with supra-additive coupling. The influence of different levels of individual neuron reliability, of recurrent and feed-forward network topologies, of dynamic connectivity (learning) and of slow dendritic spikes have to be reconsidered in this context. Our study also suggests future experiments on the propagation of synchrony due to nonlinear dendritic interactions e.g. in cultured neurons

We considered networks of

The parameters used in the given examples are

Network simulations were done in phase representation

The numerical simulations were implemented using an event based algorithm which may be outlined as follows

For the spike-train analysis, propagating chains initiated at some time

For

We implemented the network dynamics simulations in C and embedded them with MathLink into Mathematica. We used Mathematica to implement user interfaces, control programs and data analysis.

We computed the transition probabilities for the group-sizes analytically and semi-analytically. In the analytical approach, the probability distribution for the membrane potentials

Distribution of sizes of synchronous pulses in the background activity, where spikes belonging to the externally initiated propagating chain of pulses have been removed. The distributions are similar in linearly (a) and in nonlinearly (b) coupled networks. The figure exemplarily displays the sizes of spontaneously synchronized pulses in the background activity within the interval

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Tabular description of our model following ref.

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We thank M. Both, A. Draguhn, K. Gansel, T. Geisel, M. Herrmann, S. Jahnke, N. Maier, A. Morrison, S. Reichinnek, J. Schiller, D. Schmitz, W. Singer and F. Wolf for fruitful discussions.