Mean-field models for heterogeneous networks of two-dimensional integrate and fire neurons

We analytically derive mean-field models for all-to-all coupled networks of heterogeneous, adapting, two-dimensional integrate and fire neurons. The class of models we consider includes the Izhikevich, adaptive exponential and quartic integrate and fire models. The heterogeneity in the parameters leads to different moment closure assumptions that can be made in the derivation of the mean-field model from the population density equation for the large network. Three different moment closure assumptions lead to three different mean-field systems. These systems can be used for distinct purposes such as bifurcation analysis of the large networks, prediction of steady state firing rate distributions, parameter estimation for actual neurons and faster exploration of the parameter space. We use the mean-field systems to analyze adaptation induced bursting under realistic sources of heterogeneity in multiple parameters. Our analysis demonstrates that the presence of heterogeneity causes the Hopf bifurcation associated with the emergence of bursting to change from sub-critical to super-critical. This is confirmed with numerical simulations of the full network for biologically reasonable parameter values. This change decreases the plausibility of adaptation being the cause of bursting in hippocampal area CA3, an area with a sizable population of heavily coupled, strongly adapting neurons.Natural Sciences and Engineering Research Council of CanadaOntario Graduate Scholarship progra

Mean field analysis gives accurate predictions of the behaviour of large networks of sparsely coupled and heterogeneous neurons

Large networks of integrate-and-fire (IF) model neurons are often used to simulate and study the behaviour of biologically realistic networks. However, to fully study the large network behaviour requires an exploration of large regions of a multidimensional parameter space. Such exploration is generally not feasible with large network models, due to the computational time required to simulate a network with biologically significant size. To circumvent these difficulties we use a mean-field approach, based on the work of [1]. We consider a sparsely coupled, excitatory network of 10,000 Izhikevich model neurons [2], with Destexhe-type synapses [3]. The cellular models were fit to hippocampal CA1 pyramidal neurons and have heterogeneous applied currents with a normal distribution. We derived a mean-field system for the network which consists of differential equations for the mean of the adaptation current and the synaptic conductance. As CA1 is an area that displays prominent theta oscillations [4], we used the mean-field system to study how the frequency of bursting depends on various model parameters. Figure 1A shows an example study. These studies were successful in guiding numerical simulations of the large network. When parameter values determined from the mean-field analysis are used in a large network simulation, bursting of the predicted frequency occurs

Toward a multiscale modeling framework for understanding serotonergic function

Despite its importance in regulating emotion and mental wellbeing, the complex structure and function of the serotonergic system present formidable challenges toward understanding its mechanisms. In this paper, we review studies investigating the interactions between serotonergic and related brain systems and their behavior at multiple scales, with a focus on biologically-based computational modeling. We first discuss serotonergic intracellular signaling and neuronal excitability, followed by neuronal circuit and systems levels. At each level of organization, we will discuss the experimental work accompanied by related computational modeling work. We then suggest that a multiscale modeling approach that integrates the various levels of neurobiological organization could potentially transform the way we understand the complex functions associated with serotonin

Virtual deep brain stimulation: Multiscale co-simulation of a spiking basal ganglia model and a whole-brain mean-field model with The Virtual Brain

Deep brain stimulation (DBS) has been successfully applied in various neurodegenerative diseases as an effective symptomatic treatment. However, its mechanisms of action within the brain network are still poorly understood. Many virtual DBS models analyze a subnetwork around the basal ganglia and its dynamics as a spiking network with their details validated by experimental data. However, connectomic evidence shows widespread effects of DBS affecting many different cortical and subcortical areas. From a clinical perspective, various effects of DBS besides the motoric impact have been demonstrated. The neuroinformatics platform The Virtual Brain (TVB) offers a modeling framework allowing us to virtually perform stimulation, including DBS, and forecast the outcome from a dynamic systems perspective prior to invasive surgery with DBS lead placement. For an accurate prediction of the effects of DBS, we implement a detailed spiking model of the basal ganglia, which we combine with TVB via our previously developed co-simulation environment. This multiscale co-simulation approach builds on the extensive previous literature of spiking models of the basal ganglia while simultaneously offering a whole-brain perspective on widespread effects of the stimulation going beyond the motor circuit. In the first demonstration of our model, we show that virtual DBS can move the firing rates of a Parkinson's disease patient's thalamus - basal ganglia network towards the healthy regime while, at the same time, altering the activity in distributed cortical regions with a pronounced effect in frontal regions. Thus, we provide proof of concept for virtual DBS in a co-simulation environment with TVB. The developed modeling approach has the potential to optimize DBS lead placement and configuration and forecast the success of DBS treatment for individual patients

Mean-Field Models for Heterogeneous Networks of Two-Dimensional Integrate and Fire Neurons

We analytically derive mean-field models for all-to-all coupled networks of heterogeneous, adapting, two-dimensional integrate and fire neurons. The class of models we consider includes the Izhikevich, adaptive exponential, and quartic integrate and fire models. The heterogeneity in the parameters leads to different moment closure assumptions that can be made in the derivation of the mean-field model from the population density equation for the large network. Three different moment closure assumptions lead to three different mean-field systems. These systems can be used for distinct purposes such as bifurcation analysis of the large networks, prediction of steady state firing rate distributions, parameter estimation for actual neurons, and faster exploration of the parameter space. We use the mean-field systems to analyze adaptation induced bursting under realistic sources of heterogeneity in multiple parameters. Our analysis demonstrates that the presenceof heterogeneity causes the Hopf bifurcation associated with the emergence of bursting to change from sub-critical to super-critical. This is confirmed with numerical simulations of the full network for biologically reasonable parameter values. This change decreases the plausibility of adaptation being the cause of bursting in hippocampal area CA3, an area with a sizable population of heavily coupled, strongly adapting neurons

Experimentally verified reduced models of neocortical pyramidal cells

Reduced neuron models are essential tools in computational neuroscience to aid understanding from the single cell to network level. In this thesis I use these models to address two key challenges: introducing experimentally verifi�ed heterogeneity into neocortical network models, and furthering understanding of post-spike refractory mechanisms. Neocortical network models are increasingly including cell class diversity. However, within these classes significant heterogeneity is displayed, an aspect often neglected in modelling studies due to the lack of empirical constraints on the variance and covariance of neuronal parameters. To address this I quantified the response of pyramidal cells in neocortical layers 2/3-5 to square-pulse and naturalistic current stimuli. I used standard and dynamic I-V protocols to measure electrophysiological parameters, a byproduct of which is the straightforward extraction of reduced neuron models. I examined the between- and within-class heterogeneity, culminating in an algorithm to generate populations of exponential integrate-and-�re (EIF) neurons adhering to the empirical marginal distributions and covariance structure. This provides a novel tool for investigating heterogeneity in neocortical network models. Spike threshold is dynamic and, on spike initiation, displays a jump and subsequent exponential decay back to baseline. I examine extensions to the EIF model that include these dynamics, fi�nding that a simple renewal process model well captures the cell's response. It has been previously noted that a two-variable EIF model describing the voltage and threshold dynamics can be reduced to a single-variable system when the membrane and threshold time constants are similar. I examine the response properties of networks of these models by taking a perturbative approach to solving the corresponding Fokker-Planck equation, �finding the results in agreement with simulations over the physiological range of the membrane to threshold time constant ratio. Finally, I found that the observed threshold dynamics are not fully described by the inclusion of slow sodium-channel inactivation

Bifurcation Analysis of Large Networks of Neurons

The human brain contains on the order of a hundred billion neurons, each with several thousand synaptic connections. Computational neuroscience has successfully modeled both the individual neurons as various types of oscillators, in addition to the synaptic coupling between the neurons. However, employing the individual neuronal models as a large coupled network on the scale of the human brain would require massive computational and financial resources, and yet is the current undertaking of several research groups. Even if one were to successfully model such a complicated system of coupled differential equations, aside from brute force numerical simulations, little insight may be gained into how the human brain solves problems or performs tasks. Here, we introduce a tool that reduces large networks of coupled neurons to a much smaller set of differential equations that governs key statistics for the network as a whole, as opposed to tracking the individual dynamics of neurons and their connections. This approach is typically referred to as a mean-field system. As the mean-field system is derived from the original network of neurons, it is predictive for the behavior of the network as a whole and the parameters or distributions of parameters that appear in the mean-field system are identical to those of the original network. As such, bifurcation analysis is predictive for the behavior of the original network and predicts where in the parameter space the network transitions from one behavior to another. Additionally, here we show how networks of neurons can be constructed with a mean-field or macroscopic behavior that is prescribed. This occurs through an analytic extension of the Neural Engineering Framework (NEF). This can be thought of as an inverse mean-field approach, where the networks are constructed to obey prescribed dynamics as opposed to deriving the macroscopic dynamics from an underlying network. Thus, the work done here analyzes neuronal networks through both top-down and bottom-up approaches