A Complex-Valued Firing-Rate Model That Approximates the Dynamics of Spiking Networks

Abstract

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.

The receptive field is dead. Long live the receptive field?

“…However, given that it is now possible to visually identify, record from and manipulate specific cell types, the motivation to incorporate this information in models is there. The appropriate mathematics to perform the necessary reduction of such high-dimensional systems is emerging [24,25]. Studies undertaken in this spirit are beginning to address important open problems, such as the role of diverse cell types [24,26,27], pharmaceuticals [28], neuromodulation [29,30,31] and the statistics of connectivity [32,24] in shaping circuit dynamics and computation.…”

Attentional Bias Through Oscillatory Coherence Between Excitatory Activity and Inhibitory Minima

“…In that respect, an approach like the one discussed in Schaffer et al (2013) may be of interest. There, complex-valued units are related to firing rate models.…”

“…Our oscillatory network model does not have an explicit representation of spikes. It rather has the character of a complex-valued firing rate model (see Schaffer, Ostojic, & Abbott, 2013) and our related remarks in section 7. However, note that also the neurophysiological data are expressed in terms of collective quantities: averaged single-unit acitivity (SUA), multiunit activity (MUA), and local field potential (LFP).…”

Time-Variant Modeling of Brain Processes

“…The advantage of these models is that they provide a simpler description of the dynamics of a population than a larger network of spiking model neurons. These and models which are based on mean field theory [20] can be understood as neural mass models.…”

“…The adaptive MVAR model order was identified individually for each subject, resulting in values between 16 and 17. Time-variant PDC was computed by means of (10) and then averaged in the frequency bands of interest [Theta (4-7 Hz), Alpha (8-12 Hz), Beta1 (13)(14)(15)(16)(17)(18)(19)(20)(21)(22) and Beta2 (23)(24)(25)(26)(27)(28)(29)(30)]. As a result, we obtained a connectivity pattern involving the 12 ROIs and evolving in function of time.…”