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The authors have declared that no competing interests exist.

Conceived and designed the experiments: ESS SO LFA. Performed the experiments: ESS SO. Analyzed the data: ESS SO. Contributed reagents/materials/analysis tools: ESS SO LFA. Wrote the paper: ESS SO LFA.

Firing-rate models provide an attractive approach for studying large neural networks because they can be simulated rapidly and are amenable to mathematical analysis. Traditional firing-rate models assume a simple form in which the dynamics are governed by a single time constant. These models fail to replicate certain dynamic features of populations of spiking neurons, especially those involving synchronization. We present a complex-valued firing-rate model derived from an eigenfunction expansion of the Fokker-Planck equation and apply it to the linear, quadratic and exponential integrate-and-fire models. Despite being almost as simple as a traditional firing-rate description, this model can reproduce firing-rate dynamics due to partial synchronization of the action potentials in a spiking model, and it successfully predicts the transition to spike synchronization in networks of coupled excitatory and inhibitory neurons.

Neuronal responses are often characterized by the rate at which action potentials are generated rather than by the timing of individual spikes. Firing-rate descriptions of neural activity are appealing because of their comparative simplicity, but it is important to develop models that faithfully approximate dynamic features arising from spiking. In particular, synchronization or partial synchronization of spikes is an important feature that cannot be described by typical firing-rate models. Here we develop a model that is nearly as simple as the simplest firing-rate models and yet can account for a number of aspects of spiking dynamics, including partial synchrony. The model matches the dynamic activity of networks of spiking neurons with surprising accuracy. By expanding the range of dynamic phenomena that can be described by simple firing-rate equations, this model should be useful in guiding intuition about and understanding of neural circuit function.

Descriptions of neuronal spiking in terms of firing rates are widely used for both data analysis and modeling. A firing-rate description of neural data is appealing because it is much simpler than the full raster of spikes from which it is derived. In much the same spirit, firing-rate models are useful because they provide a simpler description of neural dynamics than a large network of spiking model neurons. Although firing rates are, at best, an approximation of spiking activity, they are often a sufficient description to gain insight into how neural circuits operate. Toward these approaches, it is important to develop firing-rate models that capture as much of the dynamics of spiking networks as possible.

A number of attempts have been made to derive firing-rate models as approximations to the dynamics of a population of spiking neurons

The subtlety in constructing a firing-rate model arises in trying to describe dynamics; attempting to do so leads to two questions. First, what are the dynamics of

To evaluate the validity of such a rate model, an appropriate basis of comparison is the firing rate of a population of identical spiking neurons, all receiving the same common input

A powerful method for analyzing spiking dynamics is to use the Fokker-Planck equation to compute the probability density of membrane potential values for a population of model neurons. This approach has been used to analyze the synchrony effects we consider

Classic rate models fail to describe neuronal firing when noise is insufficient to eliminate spike synchronization. The basic problem is that firing-rate dynamics are not purely exponential with a constant decay rate. The decay rate

As outlined in the

The spiking neuron models we study are all based on the equation

Firing-rate models attempt to characterize the action potentials generated by a population of spiking neurons without accounting in any way for further biophysical quantities such as the membrane potentials of the neurons. An alternative approach is to use the Fokker-Planck equation to compute the distribution of membrane potential values across the population as a function of time, and then to derive the firing rate from this distribution. This can be done by expanding the distribution in a series of modes that are eigenfunctions of the Fokker-Planck operator. In the

The computations of the dominant nonzero eigenvalue of the Fokker-Planck operator for these models are described in the Methods, and the results are shown in

The first thing apparent in

The dependence of the real part of

In evaluating differences between the integrate-and-fire models and the accuracy of our fits (

The results summarized in

Classic firing-rate models are completely specified by the function

We have already shown in

We compare the firing rate of a population of either EIF, LIF, or QIF neurons to the classic and complex-valued rate models responding to an input of the form of

The response of each rate model is compared to a spiking population receiving an input with fluctuating common term and constant variance. The common input is composed of a baseline level and a fluctuating component composed of equal-amplitude sinusoidal oscillations with random phases and frequencies of 61, 50, 33, 13.1, and 7.9 Hz.

We quantify the agreement between the activity of the spiking models and the complex-valued rate model using a shifted correlation coefficient. This is based on computing the cross-correlation between the firing rate of an integrate-and-fire population and that for the complex-valued rate model, but we allow for a small shift between the times at which these two rates are compared. As stated previously, we are primarily interested in matching dynamics over relatively slow timescales. Because of this, small temporal shifts are inconsequential. We therefore compute the correlation coefficient between these two rates at the shift that maximizes it.

In the previous section, we suggested that the better performance of the complex-valued rate model compared to the classic model is due to its ability to capture resonant behavior in the underlying integrate-and-fire model dynamics. To study this further, we computed the linear response properties of the three integrate-and-fire models and compared them to the linear response of the complex-valued rate model. In particular, we considered the responses of these models to an oscillating common input

The linear response of the complex-valued rate model is given by (

At sufficiently low noise levels, all three spiking integrate-and-fire models have a resonance at a frequency equal to their steady-state firing rate (

For the complex-valued rate model, the frequency dependence of the linear response to

As shown in the insets of

Thus far, we have shown that the complex-valued rate model can reproduce the responses of uncoupled populations of spiking neurons, but the real interest is, of course, in coupled networks. To extend our results to this case, we consider two populations of neurons, one excitatory and one inhibitory. Networks of excitatory and inhibitory neurons have been a fruitful focus of study in both rate

The spiking networks we study are large, randomly-connected networks of

We describe each population of the excitatory-inhibitory spiking network by one complex-valued rate model given by

The firing rates of the excitatory population for each of the integrate-and-fire model types (QIF, EIF, and LIF) are shown in

The stability of the asynchronous state, which is the state with constant firing rates in the complex-valued model, can be computed analytically using standard procedures (

The oscillations seen within the orange regions in

Given that the oscillations we report are not due to excitatory-inhibitory alternation, we might ask whether inhibition is needed at all. Indeed, the exponential and linear integrate-and-fire models can oscillate when the inhibitory weight is 0 (

We have presented a simple firing-rate model that captures effects caused by synchrony in networks of spiking neurons and provides a general framework to describe neural dynamics. The model, which applies generally to the class of integrate-and-fire-type spiking models, is based on a two-mode approximation of the Fokker-Planck equation. A number of researchers

A number of limitations of our model should be acknowledged. First, the model is only valid in the range of high input rates where the Fokker-Planck approach is applicable. Second, we have ignored synaptic dynamics, which can certainly play an important role in the dynamics of network firing rates

The range of noise values over which our complex-valued rate model performs well depends on whether one desires quantitative or qualitative accuracy. Higher noise in the input to a spiking population tends to lead to smoother dynamics, which are more easily matched by any firing-rate model, and our model is no exception. However, as described above, smoother dynamics also tend to have non-negligible contributions from a larger number of modes, leading to a degradation in the quantitative accuracy of our model. Summarizing these constraints, the complex-valued rate model achieves quantitative accuracy describing spiking dynamics with CVs between approximately 0.1 and 0.7. For qualitative accuracy, on the other hand, the model performs well for any CV greater than 0.1.

The two key novel aspects of the complex-valued rate model are that it relates to spiking models in a general model-independent manner, and that it continues to perform well in the low-noise regime. The generality of

The membrane potential probability density

The firing rate

After crossing threshold, the membrane potential is reset to

The threshold is an absorbing barrier, so

To begin, we consider the case when

The firing rate, expanded in terms of these eigenfunctions, is

We derived

The parameters used for each of the three neuron models considered are listed in

QIF | EIF | LIF | |

20 | |||

0 | 10 | - | |

- | 0 | 0 | |

3 | 10 | ||

10 | 10 | 10 | |

0 | 2 | 0 | |

10 | 1 | - |

For the EIF model, we compute

To compute the eigenvalues of the QIF, we take advantage of the fact that this model can be transformed into a phase model known as the theta model

We define the optimal

For the LIF, QIF and EIF models, the linear response can be computed using the methods described in Brunel & Hakim

The linear response of the complex-valued rate model can be computed by separating the complex firing rate into its real and imaginary components,

Stable asynchrony in the spiking network is analogous to a stable fixed point in the firing rate network and, similarly, stable synchrony in the spiking network is analogous to a stable limit cycle in the firing rate network. We calculated the stability of the fixed point in the firing-rate network. Instability of the fixed point results in the system finding a limit cycle.

Stability of a fixed point in the rate model is assessed by linearizing the dynamics around this fixed point. This involves taking derivatives of the right side of

The requirement for stability is that the real part of the eigenvalues of the Jacobian matrix are negative. These eigenvalues can be computed easily for a given choice of parameters, yielding a stability diagram of stable and unstable parameter regimes, separated by a bifurcation line where the real part of either eigenvalue becomes positive.

We thank M. Mattia for interesting comments and for sharing with us his unpublished ideas on a related simplification of the Fokker-Planck equation.