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DM and MT conceived and designed the experiments, performed the experiments, analyzed the data, contributed reagents/materials/analysis tools, and wrote the paper.

¤ Current address: Departement de Biologie, INSERM 497, Ecole Normale Supérieure, Paris, France

The authors have declared that no competing interests exist.

The cerebral cortex is continuously active in the absence of external stimuli. An example of this spontaneous activity is the voltage transition between an Up and a Down state, observed simultaneously at individual neurons. Since this phenomenon could be of critical importance for working memory and attention, its explanation could reveal some fundamental properties of cortical organization. To identify a possible scenario for the dynamics of Up–Down states, we analyze a reduced stochastic dynamical system that models an interconnected network of excitatory neurons with activity-dependent synaptic depression. The model reveals that when the total synaptic connection strength exceeds a certain threshold, the phase space of the dynamical system contains two attractors, interpreted as Up and Down states. In that case, synaptic noise causes transitions between the states. Moreover, an external stimulation producing a depolarization increases the time spent in the Up state, as observed experimentally. We therefore propose that the existence of Up–Down states is a fundamental and inherent property of a noisy neural ensemble with sufficiently strong synaptic connections.

The cerebral cortex is continuously active in the absence of sensory stimuli. An example of this spontaneous activity is the phenomenon of voltage transitions between two distinct levels, called Up and Down states, observed simultaneously when recoding from many neurons. This phenomenon could be of a critical importance for working memory and attention. Thus, uncovering its biological mechanism could reveal fundamental properties of the cortical organization. In this theoretical contribution, Holcman and Tsodyks propose a mathematical model of cortical dynamics that exhibits spontaneous transitions between Up and Down states. The model describes an activity of a network of interconnected neurons. A crucial component of the model is synaptic depression of interneuronal connections, which is a well-known effect that characterizes many types of synaptic connections in the cortex. Despite its simplicity, the model reproduces many properties of Up–Down transitions that were experimentally observed, and makes several intriguing predictions for future experiments. In particular, the model predicts that the time that a network spends in the Up state is highly variable, changing from a fraction of a second to more than ten seconds, which could have some interesting implications for the temporal characteristics of working memory.

In the absence of sensory inputs, cortical neural networks can exhibit complex patterns of intrinsic activity. The origin of this spontaneous activity in the cortex is still unclear, but recent studies have reported some important characteristics of the dynamics. For example, it has been demonstrated [

In previous models based on [^{+} channels. In [_{K−NA}

We propose here to investigate the role of the spontaneous activity in the generation of the Up and Down state transitions, using a modeling approach. We attribute the spontaneous fluctuations to the external noise at the level of neural populations. We demonstrate that in a sufficiently interconnected network which shows activity-dependent synaptic depression, two nonsymmetric attractors emerge. Due to the particular geometry of those attractors, the noise generates a bistability regime (two-state regime) but this is achieved in a very different way, compared with the case of symmetric attractors. When inhibitory connections are included into the model, the results are not changed qualitatively, thus to extract the main ingredients of the Up–Down states phenomena, we have decided to restrict the present analysis to a model containing only excitatory connections. Due to the activity-dependent synaptic depression, which was observed in neocortical slices [

Our analysis reveals that for certain values of the parameters of the model that characterize the strength of network connectivity, the phase space of the network contains two stable fixed points (attractors) that correspond to the stable stationary values of activity and depression of the network. One of the attractors corresponds to the state of zero activity, and the other one corresponds to higher activity. The basin of attraction associated with the first fixed point (all the initial values of the variables that dynamically flow to this fixed point) corresponds to the Down state of activity, and the basin of attraction of the second fixed point corresponds to the Up state of activity. Noise activity produces random transition between the basins, which generates the Up–Down state dynamics. The model predicts in particular that transition to the Up state is always associated with a spike generated in the network population.

The analysis of the deterministic autonomous dynamical system 1 described in Materials and Methods (when σ = 0 and _{T}_{T}_{T}_{c}_{c}_{T}_{2}. Since the dynamical system is smooth near the Up state attractor, the existence of the limit cycle _{1} = (0,1) is always an attractor of the system. To study the role of noise in a depressing neural network, the equations in Equation 1 are analyzed using the phase diagram of the system, represented in _{1}. This phenomena is the well-known exit problem for a dynamical system perturbed by random noise in general [_{r}≫τ

(A) The depression _{1} (the Down state), the saddle point _{3}_{2} (the Up state), separated by an unstable limit cycle C (dashed line). The region inside the unstable limit cycle is the basin of attraction of _{2}. By definition, it is the Up state. For a specific value of the parameter, an homoclinic curve can appear. The unstable branch of the separatrix starting from _{3} terminates inside the basin of attraction of the Down state. The noise drives the dynamics beyond the separatrices (lines with arrow converging to _{3}) but a second transition is necessary for a stochastic trajectory to enter inside the Up state delimited by C. However a single noise transition is enough to destabilize the dynamics from the Up to the Down state.

(B) Example of a trajectory in the phase space.

This spike generation in the network population always occurs during a transition from a Down to an Up state, which is a prediction of our model. When the rotation is performed in the phase space, any noise fluctuation can push the dynamics inside the Up state region. The two consecutive transitions are reasonably probable because compared with the amplitude of the noise, the separatrices are located very close to the unstable limit cycle C, as can be seen from a more careful analysis of simulations (unpublished data).

_{T} > w_{c}_{c}

(A) A typical realization of Up–Down transition dynamics in the stochastic model with _{T}

Top graph: (Up) average population voltage (mV), baseline at 0 mV; (Down) plot of the depression variable

Bottom graph: Histogram of the population voltage: fraction of the time spent in a given state. Units are in ms on the

Right column graph of (A): histogram of the duration of the Up states.

(B) Effect of an external stimulation: depolarization. When a constant external input (I = 0.8) is added into the model, the neural network spends on average more time in the Up state than in Down state. In that case, the attractor point _{1} is shifted and is now closer to the separatrix.

(C) Hyperpolarization, simulated by a current injection

Values of Parameters Used in the Simulation

The time course of the voltage is comparable to the Up–Down state dynamics observed experimentally [

The transition from the Down to the Up state can be understood as follows: the noise can lead the system to cross the border between the basin of attractions (the separatrix), but not necessarily the limit cycle. In that case, a fast spiking event is generated, as can be observed in

We study the sensitivity of the Up–Down dynamics to changes in the average value of the external input. To this end, we repeated the simulations of the network dynamics for different average values of

The top graph represents the frequency of Up states as a function of Input I. Hyperpolarizing the neurons is generated for

When the total synaptic weight _{T}_{T}_{T}_{T}_{T}

(A) Effect of increasing _{T}_{T}

(B) Effect of LTD on Up–Down state dynamics_{T}

The functional significance of the spontaneous dynamics is revealed by considering the effect on the cortical response to sensory stimulation. Several experimental studies [

The average voltage response is computed. The blue curve shows the response in the Down state. The red curve shows the response in the Up state. The horizontal black bar indicates the duration of the stimulus (starting at

We conclude that the Up–Down state dynamics can be generated from a stochastic dynamical system that describes the mean activity of a recurrent excitatory network with synaptic depression. Depending on the average _{T}_{c}_{c}

As predicted in the model, the Up and Down state transition of the network activity from one basin of attraction to another is the result of the noise activity. Since even in the Down state, the network is far from equilibrium (the synapses are always depressed), the mean time the network spends in the Down state is not approximated by the mean first passage to the separatrix, starting near the attractor of the Down state, rather it is comparable to the mean time it takes for the synapses to recover from a certain depressed activity. This deterministic relaxation is not much affected by how strong the noise is. In the Down state, the synaptic depression fluctuates much more than in the Up state. As long as the network is in the Down state, the depression recovers exponentially, until a transition in the Up state is produced.

A transition from the Up to the Down state is achieved when the noise pushes the dynamics outside the Up state region. The mean time spent in the Up state is the mean first passage time from any point inside of the basin of attraction to the unstable limit cycle and is exponentially long as a function of the amplitude of the noise [

When the noise amplitude is increased, the time spent in the Up state decreases and more transition between states occurs. As a consequence, the Up states occurring in slices (where the noise amplitude is supposed to be less than in vivo) should last longer than in vivo.

The present model does not include any channel activity, which affects the dynamics of the membrane potential. Activation of ^{+}

Analysis of

The assumption that synaptic depression can be a possible mechanism underlying the Up and Down state dynamics has already been suggested in [_{T}

Because in the Up state the response of the network to external stimulus is weaker compared with the Down state (see

This transition process can be compared with the phenomena of adaptation in cone photoreceptors, appearing when light is increased by several orders of magnitude: cone photoreceptors can continue to modulate light, where usually without adaptation the photoresponse would have been negligible. Conceivably, the dynamics associated with synaptic depression in the Up state is a mechanism of adaptation and has a global effect on the voltage sensitive channels, regulating the firing rate in the Up state.

In [

We consider a neural network connected by excitatory connections, described by the mean firing rate, which is the average of the firing rate over the population. We do not present here the equations where inhibition connections are added to the model. The change of the mean activity is modeled as a stochastic process. Neurons are connected with depressing synapses, and the state of a synapse is described by the depression parameter

where _{T}_{γ}

where _{r}

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We would like to thank R. Cossart, I. Lampl, and B. Gutkin for fruitful discussions and for their comments on the manuscript.

long-term potentiation