Mean Field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons

We derive the mean-field equations arising as the limit of a network of interacting spiking neurons, as the number of neurons goes to infinity. The neurons belong to a fixed number of populations and are represented either by the Hodgkin-Huxley model or by one of its simplified version, the Fitzhugh-Nagumo model. The synapses between neurons are either electrical or chemical. The network is assumed to be fully connected. The maximum conductances vary randomly. Under the condition that all neurons initial conditions are drawn independently from the same law that depends only on the population they belong to, we prove that a propagation of chaos phenomenon takes places, namely that in the mean-field limit, any finite number of neurons become independent and, within each population, have the same probability distribution. This probability distribution is solution of a set of implicit equations, either nonlinear stochastic differential equations resembling the McKean-Vlasov equations, or non-local partial differential equations resembling the McKean-Vlasov-Fokker- Planck equations. We prove the well-posedness of these equations, i.e. the existence and uniqueness of a solution. We also show the results of some preliminary numerical experiments that indicate that the mean-field equations are a good representation of the mean activity of a finite size network, even for modest sizes. These experiment also indicate that the McKean-Vlasov-Fokker- Planck equations may be a good way to understand the mean-field dynamics through, e.g., a bifurcation analysis.Comment: 55 pages, 9 figure

Dynamics and precursor signs for phase transitions in neural systems

This thesis investigates neural state transitions associated with sleep, seizure and anaesthesia. The aim is to address the question: How does a brain traverse the critical threshold between distinct cortical states, both healthy and pathological? Specifically we are interested in sub-threshold neural behaviour immediately prior to state transition. We use theoretical neural modelling (single spiking neurons, a network of these, and a mean-field continuum limit) and in vitro experiments to address this question. Dynamically realistic equations of motion for thalamic relay neuron, reticular nuclei, cortical pyramidal and cortical interneuron in different vigilance states are developed, based on the Izhikevich spiking neuron model. A network of cortical neurons is assembled to examine the behaviour of the gamma-producing cortical network and its transition to lower frequencies due to effect of anaesthesia. Then a three-neuron model for the thalamocortical loop for sleep spindles is presented. Numerical simulations of these networks confirms spiking consistent with reported in vivo measurement results, and provides supporting evidence for precursor indicators of imminent phase transition due to occurrence of individual spindles. To complement the spiking neuron networks, we study the Wilson–Cowan neural mass equations describing homogeneous cortical columns and a 1D spatial cluster of such columns. The abstract representation of cortical tissue by a pair of coupled integro-differential equations permits thorough linear stability, phase plane and bifurcation analyses. This model shows a rich set of spatial and temporal bifurcations marking the boundary to state transitions: saddle-node, Hopf, Turing, and mixed Hopf–Turing. Close to state transition, white-noise-induced subthreshold fluctuations show clear signs of critical slowing down with prolongation and strengthening of autocorrelations, both in time and space, irrespective of bifurcation type. Attempts at in vitro capture of these predicted leading indicators form the last part of the thesis. We recorded local field potentials (LFPs) from cortical and hippocampal slices of mouse brain. State transition is marked by the emergence and cessation of spontaneous seizure-like events (SLEs) induced by bathing the slices in an artificial cerebral spinal fluid containing no magnesium ions. Phase-plane analysis of the LFP time-series suggests that distinct bifurcation classes can be responsible for state change to seizure. Increased variance and growth of spectral power at low frequencies (f < 15 Hz) was observed in LFP recordings prior to initiation of some SLEs. In addition we demonstrated prolongation of electrically evoked potentials in cortical tissue, while forwarding the slice to a seizing regime. The results offer the possibility of capturing leading temporal indicators prior to seizure generation, with potential consequences for understanding epileptogenesis. Guided by dynamical systems theory this thesis captures evidence for precursor signs of phase transitions in neural systems using mathematical and computer-based modelling as well as in vitro experiments

Gain control with A-type potassium current: IA as a switch between divisive and subtractive inhibition

Neurons process information by transforming barrages of synaptic inputs into spiking activity. Synaptic inhibition suppresses the output firing activity of a neuron, and is commonly classified as having a subtractive or divisive effect on a neuron's output firing activity. Subtractive inhibition can narrow the range of inputs that evoke spiking activity by eliminating responses to non-preferred inputs. Divisive inhibition is a form of gain control: it modifies firing rates while preserving the range of inputs that evoke firing activity. Since these two "modes" of inhibition have distinct impacts on neural coding, it is important to understand the biophysical mechanisms that distinguish these response profiles. We use simulations and mathematical analysis of a neuron model to find the specific conditions for which inhibitory inputs have subtractive or divisive effects. We identify a novel role for the A-type Potassium current (IA). In our model, this fast-activating, slowly- inactivating outward current acts as a switch between subtractive and divisive inhibition. If IA is strong (large maximal conductance) and fast (activates on a time-scale similar to spike initiation), then inhibition has a subtractive effect on neural firing. In contrast, if IA is weak or insufficiently fast-activating, then inhibition has a divisive effect on neural firing. We explain these findings using dynamical systems methods to define how a spike threshold condition depends on synaptic inputs and IA. Our findings suggest that neurons can "self-regulate" the gain control effects of inhibition via combinations of synaptic plasticity and/or modulation of the conductance and kinetics of A-type Potassium channels. This novel role for IA would add flexibility to neurons and networks, and may relate to recent observations of divisive inhibitory effects on neurons in the nucleus of the solitary tract.Comment: 20 pages, 11 figure

Recent advances in mathematical modeling and statistical analysis of exocytosis in endocrine cells

open5noMost endocrine cells secrete hormones as a result of Ca(2+)-regulated exocytosis, i.e., fusion of the membranes of hormone-containing secretory granules with the cell membrane, which allows the hormone molecules to escape to the extracellular space. As in neurons, electrical activity and cell depolarization open voltage-sensitive Ca(2+) channels, and the resulting Ca(2+) influx elevate the intracellular Ca(2+) concentration, which in turn causes exocytosis. Whereas the main molecular components involved in exocytosis are increasingly well understood, quantitative understanding of the dynamical aspects of exocytosis is still lacking. Due to the nontrivial spatiotemporal Ca(2+) dynamics, which depends on the particular pattern of electrical activity as well as Ca(2+) channel kinetics, exocytosis is dependent on the spatial arrangement of Ca(2+) channels and secretory granules. For example, the creation of local Ca(2+) microdomains, where the Ca(2+) concentration reaches tens of µM, are believed to be important for triggering exocytosis. Spatiotemporal simulations of buffered Ca(2+) diffusion have provided important insight into the interplay between electrical activity, Ca(2+) channel kinetics, and the location of granules and Ca(2+) channels. By confronting simulations with statistical time-to-event (or survival) regression analysis of single granule exocytosis monitored with TIRF microscopy, a direct connection between location and rate of exocytosis can be obtained at the local, single-granule level. To get insight into whole-cell secretion, simplifications of the full spatiotemporal dynamics have shown to be highly helpful. Here, we provide an overview of recent approaches and results for quantitative analysis of Ca(2+) regulated exocytosis of hormone-containing granules.openPedersen, Morten Gram; Tagliavini, Alessia; Cortese, Giuliana; Riz, Michela; Montefusco, FrancescoPedersen, MORTEN GRAM; Tagliavini, Alessia; Cortese, Giuliana; Riz, Michela; Montefusco, Francesc

Accelerating SNN Training with Stochastic Parallelizable Spiking Neurons

Spiking neural networks (SNN) are able to learn spatiotemporal features while using less energy, especially on neuromorphic hardware. The most widely used spiking neuron in deep learning is the Leaky Integrate and Fire (LIF) neuron. LIF neurons operate sequentially, however, since the computation of state at time t relies on the state at time t-1 being computed. This limitation is shared with Recurrent Neural Networks (RNN) and results in slow training on Graphics Processing Units (GPU). In this paper, we propose the Stochastic Parallelizable Spiking Neuron (SPSN) to overcome the sequential training limitation of LIF neurons. By separating the linear integration component from the non-linear spiking function, SPSN can be run in parallel over time. The proposed approach results in performance comparable with the state-of-the-art for feedforward neural networks on the Spiking Heidelberg Digits (SHD) dataset, outperforming LIF networks while training 10 times faster and outperforming non-spiking networks with the same network architecture. For longer input sequences of 10000 time-steps, we show that the proposed approach results in 4000 times faster training, thus demonstrating the potential of the proposed approach to accelerate SNN training for very large datasets

Mechanisms Leading to Rhythm Cessation in the Respiratory PreBotzinger Complex Due to Piecewise Cumulative Neuronal Deletions

The brainstem pre-Botzinger complex (preBotC) generates the rhythm underlying inspiratory breathing movements and its core interneurons are derived from Dbx-1 expressing precursors. Recurrent synaptic excitation is required to initiate inspiratory bursts, but whether excitatory synaptic mechanisms also contribute to inspiratory–expiratory phase transition is unknown. Here, we examined the role of short-term synaptic depression using a rhythmically active neonatal mouse brainstem slice preparation. We show that different axonal projections to Dbx-1 PreBotC neurons undergo activity-dependent depression and we identify a refractory period (approx. 2 s) after endogenous inspiratory bursts that precludes light-evoked bursts in channelrhodopsin-expressing Dbx1 Pre-BotC neurons. We demonstrate that the duration of the refractory period---but neither the cycle period nor the magnitude of endogenous inspiratory burst---is sensitive to changes in extracellular Ca^2+. Further, we show that postsynaptic factors are unlikely to explain the refractory period or its modulation by Ca^2+. Our findings are consistent with the hypothesis that short-term synaptic depression in Dbx-1 Pre-BotC neurons influences the inspiratory-expiratory phase transition during respiratory rhythmogenesis