364 research outputs found
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
Analysis of Nonlinear Noisy Integrate\&Fire Neuron Models: blow-up and steady states
Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks
can be written as Fokker-Planck-Kolmogorov equations on the probability density
of neurons, the main parameters in the model being the connectivity of the
network and the noise. We analyse several aspects of the NNLIF model: the
number of steady states, a priori estimates, blow-up issues and convergence
toward equilibrium in the linear case. In particular, for excitatory networks,
blow-up always occurs for initial data concentrated close to the firing
potential. These results show how critical is the balance between noise and
excitatory/inhibitory interactions to the connectivity parameter
Deterministic continutation of stochastic metastable equilibria via Lyapunov equations and ellipsoids
Numerical continuation methods for deterministic dynamical systems have been
one of the most successful tools in applied dynamical systems theory.
Continuation techniques have been employed in all branches of the natural
sciences as well as in engineering to analyze ordinary, partial and delay
differential equations. Here we show that the deterministic continuation
algorithm for equilibrium points can be extended to track information about
metastable equilibrium points of stochastic differential equations (SDEs). We
stress that we do not develop a new technical tool but that we combine results
and methods from probability theory, dynamical systems, numerical analysis,
optimization and control theory into an algorithm that augments classical
equilibrium continuation methods. In particular, we use ellipsoids defining
regions of high concentration of sample paths. It is shown that these
ellipsoids and the distances between them can be efficiently calculated using
iterative methods that take advantage of the numerical continuation framework.
We apply our method to a bistable neural competition model and a classical
predator-prey system. Furthermore, we show how global assumptions on the flow
can be incorporated - if they are available - by relating numerical
continuation, Kramers' formula and Rayleigh iteration.Comment: 29 pages, 7 figures [Fig.7 reduced in quality due to arXiv size
restrictions]; v2 - added Section 9 on Kramers' formula, additional
computations, corrected typos, improved explanation
Adaptive Parameter Selection for Deep Brain Stimulation in Parkinson’s Disease
Each year, around 60,000 people are diagnosed with Parkinson’s disease (PD) and the economic burden of PD is at least 12,000 to $6,000 per year with the addition of neuromodulation therapies such as Deep Brain Stimulation (DBS), transcranial Direct Current Stimulation (tDCS), Transcranial Magnetic Stimulation (TMS), etc. In neurodegenerative disorders such as PD, deep brain stimulation (DBS) is a desirable approach when the medication is less effective for treating the symptoms. DBS incorporates transferring electrical pulses to a specific tissue of the central nervous system and obtaining therapeutic results by modulating the neuronal activity of that region. The hyperkinetic symptoms of PD are associated with the ensembles of interacting oscillators that cause excess or abnormal synchronous behavior within the Basal Ganglia (BG) circuitry. Delayed feedback stimulation is a closed loop technique shown to suppress this synchronous oscillatory activity. Deep Brain Stimulation via delayed feedback is known to destabilize the complex intermittent synchronous states. Computational models of the BG network are often introduced to investigate the effect of delayed feedback high frequency stimulation on partially synchronized dynamics. In this work, we developed several computational models of four interacting nuclei of the BG as well as considering the Thalamo-Cortical local effects on the oscillatory dynamics. These models are able to capture the emergence of 34 Hz beta band oscillations seen in the Local Field Potential (LFP) recordings of the PD state. Traditional High Frequency Stimulations (HFS) has shown deficiencies such as strengthening the synchronization in case of highly fluctuating neuronal activities, increasing the energy consumed as well as the incapability of activating all neurons in a large-scale network. To overcome these drawbacks, we investigated the effects of the stimulation waveform and interphase delays on the overall efficiency and efficacy of DBS. We also propose a new feedback control variable based on the filtered and linearly delayed LFP recordings. The proposed control variable is then used to modulate the frequency of the stimulation signal rather than its amplitude. In strongly coupled networks, oscillations reoccur as soon as the amplitude of the stimulus signal declines. Therefore, we show that maintaining a fixed amplitude and modulating the frequency might ameliorate the desynchronization process, increase the battery lifespan and activate substantial regions of the administered DBS electrode. The charge balanced stimulus pulse itself is embedded with a delay period between its charges to grant robust desynchronization with lower amplitude needed. The efficiency and efficacy of the proposed Frequency Adjustment Stimulation (FAS) protocol in a delayed feedback method might contribute to further investigation of DBS modulations aspired to address a wide range of abnormal oscillatory behaviors observed in neurological disorders. Adaptive stimulation can open doors towards simultaneous stimulation with MRI recordings. We additionally propose a new pipeline to investigate the effect of Transcranial Magnetic Stimulation (TMS) on patient specific models. The pipeline allows us to generate a full head segmentation based on each individual MRI data. In the next step, the neurosurgeon can adaptively choose the proper location of stimulation and transmit accurate magnetic field with this pipeline
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