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In this article, we present a computational study of the Conductance-Based Adaptive Exponential (CAdEx) integrate-and-fire neuronal model, focusing on its multiple timescale nature, and on how it shapes its main dynamical regimes. In particular, we show that the spiking and so-called delayed bursting regimes of the model are triggered by discontinuity-induced bifurcations that are directly related to the multiple-timescale aspect of the model, and are mediated by canard solutions. By means of a numerical bifurcation analysis of the model, using the software package coco, we can precisely describe the mechanisms behind these dynamical scenarios. Spike-increment transitions are revealed. These transitions are accompanied by a fold and a period-doubling bifurcation, and are organised in parameter space along an isola periodic solutions with resets. Finally, we also unveil the presence of a homoclinic bifurcation terminating a canard explosion which, together with the presence of resets, organises the delayed bursting regime of the model. [ABSTRACT FROM AUTHOR]

Transcranial electrical brain stimulation techniques like transcranial direct current (tDCS) and transcranial alternating current (tACS) have emerged as potential tools for treating neurological diseases by modulating cortical excitability. These techniques deliver small electric currents to the brain non-invasively through electrodes on the scalp. tDCS uses constant direct current which weakly alters the membrane voltage of cortical neurons, while tACS utilizes alternating current to target and enhance cortical oscillations, though the underlying mechanisms are not fully understood more specifically. To elucidate how tACS perturbs endogenous network dynamics, we simulated spiking neuron network models. We identified distinct roles of the depolarizing and hyperpolarizing phases in driving network activity towards and away from the strong nonlinearity provided by pyramidal neurons. Exploring resonance effects, we found matching tACS frequency to the network's endogenous resonance frequency creates greater entrainment. Based on this, we developed an algorithm to determine the network's endogenous frequency, phase, and amplitude, then deliver optimized tACS to entrain network oscillations. Together, these computational results provide mechanistic insight into the effects of tACS on network dynamics and could inform future closed-loop tACS systems that dynamically tune stimulation parameters to ongoing brain activity. [ABSTRACT FROM AUTHOR]

We investigate the response characteristics of a generalized FitzHugh-Nagumo model under an external sinusoidal current and the synchronization of two neurons coupled with a gap junction. In the autonomous case, we find analytically by the Lindsted's method that the system can admit tristable activities in silence, subthreshold, and nerve pulse; depending on the conductance parameters and the state of ionic conductance. In the presence of an external sinusoidal current, we find by numerical simulations that neurons can exhibit a coexistence between different spiking patterns and periodic waves, which are well observed in the structure of the recurrence plot. We further study the synchronization between coupled neurons each admitting bistable activities, such as a coexistence between chaotic (active) and silence (inactive) regimes. We apply recurrence analysis tool to reveal the range of the coupling parameter where synchronization occurs, as well as the dynamical transitions between the synchronous coexisting states (hysteresis phenomenon). The coupling strength is an indicator of the phenomenon of synchronization that can also bring the system to any of the desired synchronous attractors. These phenomena of synchronization and the control between synchronous states can be improved by the presence of an external electrical field. The switching of the coupled neurons to bursting patterns or to periodic waves explains the well-known properties of excitatory (switching on) or inhibitory (switching off) synaptic coupling, respectively; while the unstable signal separating the two stable synchronous signals can be taken as the synaptic threshold. Rather, this study adds to our theoretical understanding of the topic and poses new challenges for investigation. Experimental investigations are required to validate these conclusions in real-world settings, and biological implications must be evaluated within the particular framework of the modeling that was done. [ABSTRACT FROM AUTHOR]

Intrinsic bursting neurons represent a common neuronal type that displays bursting patterns upon depolarization stimulation. These neurons can be described by a system of seven-dimensional equations, which pose a challenge for dynamical analysis. To overcome this limitation, we employed the projection reduction method to reduce the dimensionality of the model. Our approach demonstrated that the reduced model retained the inherent bursting characteristics of the original model. Following reduction, we investigated the bi-parameter bifurcation of the equilibrium point in the reduced model. Specifically, we analyzed the Bogdanov-Takens bifurcation that arises in the reduced system. Notably, the topological structure of the neuronal model near the bifurcation point can be effectively revealed with our proposed method. By leveraging the proposed projection reduction method, we could explore the bursting mechanism in the reduced Pospischil model with greater precision. Our approach offers an effective foundation for generating theories and hypotheses that can be tested experimentally. Furthermore, it enables links to be drawn between neuronal morphology and function, thereby facilitating a deeper understanding of the complex dynamical behaviors that underlie intrinsic bursting neurons.

Intrinsic bursting neurons represent a common neuronal type that displays bursting patterns upon depolarization stimulation. These neurons can be described by a system of seven-dimensional equations, which pose a challenge for dynamical analysis. To overcome this limitation, we employed the projection reduction method to reduce the dimensionality of the model. Our approach demonstrated that the reduced model retained the inherent bursting characteristics of the original model. Following reduction, we investigated the bi-parameter bifurcation of the equilibrium point in the reduced model. Specifically, we analyzed the Bogdanov-Takens bifurcation that arises in the reduced system. Notably, the topological structure of the neuronal model near the bifurcation point can be effectively revealed with our proposed method. By leveraging the proposed projection reduction method, we could explore the bursting mechanism in the reduced Pospischil model with greater precision. Our approach offers an effective foundation for generating theories and hypotheses that can be tested experimentally. Furthermore, it enables links to be drawn between neuronal morphology and function, thereby facilitating a deeper understanding of the complex dynamical behaviors that underlie intrinsic bursting neurons. [ABSTRACT FROM AUTHOR]

Hybrid synapses widely exist in the brain neural system, but how memristive and plastic chemical synapses cooperatively modulate the collective dynamics of neurons remains largely unknown. Here, we constructed self-organized networks with two heterogeneous FitzHugh–Nagumo (FHN) neurons coupled with memristive and chemical synapses, wherein the chemical synapse is modulated by the spike-timing-dependent plasticity (STDP) rule. Additionally, three kinds of network models involving excitatory–excitatory (E – E) neurons, high excitatory–inhibitory (high E – I) neurons and low excitatory–inhibitory (low E – I) neurons were constructed. The modulation of memristive synapses on the structure and dynamics of self-organized neuronal networks is greatly dependent on model selection. Stronger coupling of memristive synapses induces consistently more stable network structure and enhanced network synchronization in the E – E and high E – I models but has complex effects on the low E – I neuronal network. In contrast, increasing the closing rate of memristive synapses has little effect on the E – E and high E – I networks but can accelerate the self-organization process and result in more complex firing patterns and weaker synchronization in the low E – I network. These results provide further understanding of the mechanism of the self-organized neuronal network dynamics underlying hybrid synapses and neuronal excitation. [ABSTRACT FROM AUTHOR]

In order to solve the chaotic synchronization problem of two neuron models with different structures, the dynamical behaviours of Fitzhugh-Nagumo (FHN) neurons and Morris-Lecar (ML) neurons are studied respectively. Based on Lyapunov stability theory, a controller is designed by using the method of projection synchronization, and the chaotic synchronization of two neuron models with different structures is realized by the controller. The neural synchronization system is applied to chaotic secure communication, and the simulation results verifies its effectiveness and reliability in secure communication.

In order to solve the chaotic synchronization problem of two neuron models with different structures, the dynamical behaviours of Fitzhugh-Nagumo (FHN) neurons and Morris-Lecar (ML) neurons are studied respectively. Based on Lyapunov tability theory, a controller is designed by using the method of projection synchronization, and the chaotic synchronization of wo neuron models with different structures is realized by the controller. The neural synchronization system is applied to chaotic secure communication, and the simulation results verifies its effectiveness and reliability in secure communication. [ABSTRACT FROM AUTHOR]

This paper presents a higher order uniformly convergent discretization for second order singularly perturbed parabolic differential equation with delay and advance terms. The retarded terms are approximated by the Taylor series which give rise to a nearby singularly perturbed parabolic differential equation. A hybrid numerical algorithm based on implicit Euler scheme is proposed to discretize the time variable and a combined finite difference scheme made out of modified upwind and central difference schemes to discretize the spatial variable on a piecewise uniform mesh of Shishkin type in space and a uniform mesh in time. The existence and uniqueness of a solution for the proposed hybrid algorithm is analyzed. It is proved that the algorithm is ϵ -uniformly convergent of almost second order in space, O (N - 2 (ln N) 2) and first order in time, O (M - 1) . A few numerical experiments supporting the results are presented. The efficiency of the proposed hybrid algorithm is demonstrated by comparing upwind and modified upwind algorithm. [ABSTRACT FROM AUTHOR]

Computational models in neuroscience can be used to predict causal relationships between biological mechanisms in neurons and networks, such as the effect of blocking an ion channel or synaptic connection on neuron activity. Since developing a biophysically realistic, single neuron model is exceedingly difficult, software has been developed for automatically adjusting parameters of computational neuronal models. The ideal optimization software should work with commonly used neural simulation software; thus, we present software which works with models specified in declarative format for the MOOSE simulator. Experimental data can be specified using one of two different file formats. The fitness function is customizable as a weighted combination of feature differences. The optimization itself uses the covariance matrix adaptation-evolutionary strategy, because it is robust in the face of local fluctuations of the fitness function, and deals well with a high-dimensional and discontinuous fitness landscape. We demonstrate the versatility of the software by creating several model examples of each of four types of neurons (two subtypes of spiny projection neurons and two subtypes of globus pallidus neurons) by tuning to current clamp data. Optimizations reached convergence within 1,600–4,000 model evaluations (200–500 generations × population size of 8). Analysis of the parameters of the best fitting models revealed differences between neuron subtypes, which are consistent with prior experimental results. Overall our results suggest that this easy-to-use, automatic approach for finding neuron channel parameters may be applied to current clamp recordings from neurons exhibiting different biochemical markers to help characterize ionic differences between other neuron subtypes.

We consider the process {V (t) : t ≥ 0} defined by V (t) = v0eX(t) (for all t ≥ 0), where v0 > 0 and {X(t) : t ≥ 0} is a compound Poisson process with exponentially distributed jumps and a negative drift. This process can be seen as the neuronal membrane potential in the stochastic model for the firing activity of a neuronal unit presented in Di Crescenzo and Martinucci (Math Biosci 209(2):547-563 2007). We also consider the process {V~(t):t≥0}, where V~(t)=v0eX~(t) (for all t ≥ 0) and {X~(t):t≥0} is the Normal approximation (as t→∞) of the process {X(t) : t ≥ 0}. In this paper we are interested in the first-passage times through a constant firing threshold β (where β > v0) for both processes {V (t) : t ≥ 0} and {V~(t):t≥0}; our aim is to study their asymptotic behavior as β→∞ in the fashion of large deviations. We also study some statistical applications for both models, with some numerical evaluations and simulation results. [ABSTRACT FROM AUTHOR]

A class of time‐dependent singularly perturbed convection‐diffusion problems with retarded terms arising in computational neuroscience is considered. In particular, a numerical scheme for the parabolic convection‐diffusion problem where the second‐order derivative with respect to the spatial direction is multiplied by a small perturbation parameter ϵ 2 ;   ϵ ∈ ( 0 , 1 ] and the shifts δ ,   η are of o ( ϵ ) is constructed. The Taylor series expansion is used to tackle the shift terms. The continuous problem is semidiscretized using the Crank‐Nicolson finite difference method in the temporal direction and the resulting set of ordinary differential equations is discretized using a midpoint upwind finite difference scheme on an appropriate piecewise uniform mesh, which is dense in the boundary layer region. It is shown that the proposed numerical scheme is second‐order accurate in time and almost first‐order accurate in space with respect to the perturbation parameter ϵ . To validate the computational results and efficiency of the method some numerical examples are encountered and the numerical results are compared with some existing results. It is observed that the numerical approximations are fairly good irrespective of the size of the delay and the advance till they are of o ( ϵ ) . The effect of the shifts on the boundary layer has also been observed. [ABSTRACT FROM AUTHOR]

Computational models in neuroscience can be used to predict causal relationships between biological mechanisms in neurons and networks, such as the effect of blocking an ion channel or synaptic connection on neuron activity. Since developing a biophysically realistic, single neuron model is exceedingly difficult, software has been developed for automatically adjusting parameters of computational neuronal models. The ideal optimization software should work with commonly used neural simulation software; thus, we present software which works with models specified in declarative format for the MOOSE simulator. Experimental data can be specified using one of two different file formats. The fitness function is customizable as a weighted combination of feature differences. The optimization itself uses the covariance matrix adaptation-evolutionary strategy, because it is robust in the face of local fluctuations of the fitness function, and deals well with a high-dimensional and discontinuous fitness landscape. We demonstrate the versatility of the software by creating several model examples of each of four types of neurons (two subtypes of spiny projection neurons and two subtypes of globus pallidus neurons) by tuning to current clamp data. Optimizations reached convergence within 1,600–4,000 model evaluations (200–500 generations × population size of 8). Analysis of the parameters of the best fitting models revealed differences between neuron subtypes, which are consistent with prior experimental results. Overall our results suggest that this easy-to-use, automatic approach for finding neuron channel parameters may be applied to current clamp recordings from neurons exhibiting different biochemical markers to help characterize ionic differences between other neuron subtypes. [ABSTRACT FROM AUTHOR]

Rubinstein-Taybi syndrome (RSTS) is a rare neurodevelopmental disorder characterized by distinctive facial features, growth retardation, broad thumbs and toes and mild to severe intellectual disability, caused by heterozygous mutations in either CREBBP or EP300 genes, encoding the homologous CBP and p300 lysine-acetyltransferases and transcriptional coactivators. No RSTS in vitro induced Pluripotent Stem Cell (iPSC)-neuronal model is available yet to achieve mechanistic insights on cognitive impairment of RSTS patients. We established iPSC-derived neurons (i-neurons) from peripheral blood cells of three CREBBP - and two EP300 -mutated patients displaying different levels of intellectual disability, and four unaffected controls. Pan neuronal and cortical-specific markers were expressed by all patients' i-neurons. Altered morphology of patients' differentiating neurons, showing reduced branch length and increased branch number, and hypoexcitability of differentiated neurons emerged as potential disease biomarkers. Anomalous neuronal morphology and reduced excitability varied across different RSTS patients' i-neurons. Further studies are needed to validate these markers and assess whether they reflect cognitive and behavioural impairment of the donor patients. [ABSTRACT FROM AUTHOR]

Dynamic causal modeling (DCM) is a methodological approach to study effective connectivity among brain regions. Based on a set of observations and a biophysical model of brain interactions, DCM uses a Bayesian framework to estimate the posterior distribution of the free parameters of the model ( e.g. modulation of connectivity) and infer architectural properties of the most plausible model ( i.e. model selection). When modeling electrophysiological event-related responses, the estimation of the model relies on the integration of the system of delay differential equations (DDEs) that describe the dynamics of the system. In this technical note, we compared two numerical schemes for the integration of DDEs. The first, and standard, scheme approximates the DDEs (more precisely, the state of the system, with respect to conduction delays among brain regions) using ordinary differential equations (ODEs) and solves it with a fixed step size. The second scheme uses a dedicated DDEs solver with adaptive step sizes to control error, making it theoretically more accurate. To highlight the effects of the approximation used by the first integration scheme in regard to parameter estimation and Bayesian model selection, we performed simulations of local field potentials using first, a simple model comprising 2 regions and second, a more complex model comprising 6 regions. In these simulations, the second integration scheme served as the standard to which the first one was compared. Then, the performances of the two integration schemes were directly compared by fitting a public mismatch negativity EEG dataset with different models. The simulations revealed that the use of the standard DCM integration scheme was acceptable for Bayesian model selection but underestimated the connectivity parameters and did not allow an accurate estimation of conduction delays. Fitting to empirical data showed that the models systematically obtained an increased accuracy when using the second integration scheme. We conclude that inference on connectivity strength and delay based on DCM for EEG/MEG requires an accurate integration scheme. [ABSTRACT FROM AUTHOR]

To understand how single neurons process sensory information, it is necessary to develop suitable stochastic models to describe the response variability of the recorded spike trains. Spikes in a given neuron are produced by the synergistic action of sodium and potassium of the voltage-dependent channels that open or close the gates. Hodgkin and Huxley (HH) equations describe the ionic mechanisms underlying the initiation and propagation of action potentials, through a set of nonlinear ordinary differential equations that approximate the electrical characteristics of the excitable cell. Path integral provides an adequate approach to compute quantities such as transition probabilities, and any stochastic system can be expressed in terms of this methodology. We use the technique of path integrals to determine the analytical solution driven by a non-Gaussian colored noise when considering the HH equations as a stochastic system. The different neuronal dynamics are investigated by estimating the path integral solutions driven by a non-Gaussian colored noise q . More specifically we take into account the correlational structures of the complex neuronal signals not just by estimating the transition probability associated to the Gaussian approach of the stochastic HH equations, but instead considering much more subtle processes accounting for the non-Gaussian noise that could be induced by the surrounding neural network and by feedforward correlations. This allows us to investigate the underlying dynamics of the neural system when different scenarios of noise correlations are considered. [ABSTRACT FROM AUTHOR]

Because every spike of a neuron is determined by input signals, a train of spikes may contain information about the dynamics of unobserved neurons. A state-space method based on the leaky integrate-and-fire model, describing neuronal transformation from input signals to a spike train has been proposed for tracking input parameters represented by their mean and fluctuation [11]. In the present paper, we propose to make the estimation more realistic by adopting an LIF model augmented with an adaptive moving threshold. Moreover, because the direct state-space method is computationally infeasible for a data set comprising thousands of spikes, we further develop a practical method for transforming instantaneous firing characteristics back to input parameters. The instantaneous firing characteristics, represented by the firing rate and non-Poisson irregularity, can be estimated using a computationally feasible algorithm. We applied our proposed methods to synthetic data to clarify that they perform well.

In this paper, we discuss the shape of traveling wave front solutions to a neuronal model with the connection function to be of lateral excitation type. This means that close connecting cells have an inhibitory influence, while cells that aremore distant have an excitatory influence. We give results on the shape of the wave fronts solutions, which exhibit different shapes depend ing on the size of a threshold parameter.

The mismatch negativity (MMN) is thought to be an index of the automatic activation of a specialized network for active prediction and deviance detection in the auditory cortex. It is consistently reduced in schizophrenic patients and has received a lot of interest as a clinical and translational tool. The main neuronal hypothesis regarding the mechanisms leading to a reduced MMN in schizophrenic patients is a dysfunction of NMDA receptors (NMDA-R). However, this hypothesis has never been implemented in a neuronal model. In this paper, we examine the consequences of NMDA-R dysfunction in a neuronal model of MMN based on predictive coding principle. I also investigate how predictive processes may interact with synaptic adaptation in MMN generations and examine the consequences of this interaction for the use of MMN paradigms in schizophrenia research. [ABSTRACT FROM AUTHOR]