Mathematical Models of Dynamic Behavior of Individual Neural Networks of Central Nervous System (original) (raw)
Related papers
Biological Neuronal Networks, Modeling of
In recent decades, since the seminal work of AL Hodgkin and AF Huxley (1), the study of the dynamical phenomena emerging in a network of biological neurons has been approached by means of mathematical descriptions, computer simulations (2, 3), and neuromorphic electronic hardware implementations (4). Several models1 have been proposed in the literature, and a large class of them share similar qualitative features.
Dynamics of neural systems with time delays
2017
Complex networks are ubiquitous in nature. Numerous neurological diseases, such as Alzheimer's, Parkinson's, epilepsy are caused by the abnormal collective behaviour of neurons in the brain. In particular, there is a strong evidence that Parkinson's disease is caused by the synchronisation of neurons, and understanding how and why such synchronisation occurs will bring scientists closer to the design and implementation of appropriate control to support desynchronisation required for the normal functioning of the brain. In order to study the emergence of (de)synchronisation, it is necessary first to understand how the dynamical behaviour of the system under consideration depends on the changes in systems parameters. This can be done using a powerful mathematical method, called bifurcation analysis, which allows one to identify and classify different dynamical regimes, such as, for example, stable/unstable steady states, Hopf and fold bifurcations, and find periodic soluti...
Model for a neural network structure and signal transmission
We present a model of a neural network that is based on the diffusion-limited-aggregation ~DLA! structure from fractal physics. A single neuron is one DLA cluster, while a large number of clusters, in an interconnected fashion, make up the neural network. Using simulation techniques, a signal is randomly generated and traced through its transmission inside the neuron and from neuron to neuron through the synapses. The activity of the entire neural network is monitored as a function of time. The characteristics included in the model contain, among others, the threshold for firing, the excitatory or inhibitory character of the synapse, the synaptic delay, and the refractory period. The system activity results in ‘‘noisy’’ time series that exhibit an oscillatory character. Standard power spectra are evaluated and fractal analyses performed, showing that the system is not chaotic, but the varying parameters can be associated with specific values of fractal dimensions. It is found that the network activity is not linear with the system parameters, e.g., with the numbers of active synapses. The details of this behavior may have interesting repercussions from the neurological point of view.
2004
We introduce a model for the electrical behavior of brain cells, based on a model introduced in (8). This model basically makes analogies be- tween electrical circuits and the way the body and synapse of brain cells work. Numerical simulation is implemented seeking for synchro- nization; what the numerical results show is synchronization in case of little, strong interaction (excitation),
Modèles mathématiques pour l'étude des phénomènes de synchronisation dans les réseaux neuronaux
2006
The spike train, i.e. the sequence of the action potential timings of a single unit, is the usual data that is analyzed in electrophysiological recordings for the description of the firing pattern which is supposed to characterize a certain type of cell.. We present the results obtained describing the firing activity of a small network of neurons with a mathematical jump diffusion model. That is the membrane potential as a function of time is given by the sum of a stochastic diffusion process and two counting processes that provoke jumps of constant sizes at discrete random times. Different distributions are considered for such processes. Two main results emerge. The first one is that interspike intervals (ISI) histograms show more than one peak (multimodality) and exhibit a resonant like behavior. This fact suggests that in correspondence of each mode (i.e. the lag of the maxima) the cell has a higher probability of firing such that the the lags become characteristic times of the c...
Computational framework for behavioural modelling of neural subsystems
Neurocomputing, 2009
This paper presents a new approach to the problem of modelling living system dynamics. Our point of view claims the fact that behind the biological apparent complexity, a hidden simplicity may appear when a suitable modelling is developed. The framework is inspired on the computing features of biological systems by involving a set of elementary standard behaviours that can be combined in order to emulate more complex behaviours. The algebraic formalization is based on both a recursive primitive operation defined by a table which models the elementary behaviours and a multilevel operating mode that carries out behaviour combinations. A parametric architecture implements the model, providing a good trade-off between time delay calculation and memory requirements. In this paper, the simulation of neural subsystems is considered as an application. The comparison with other simulation techniques outlines the capabilities of our method to provide an accurate modelling together with a very simple circuit implementation.
Dynamics of Neural Systems with Discrete and Distributed Time Delays
SIAM Journal on Applied Dynamical Systems, 2015
In real-world systems, interactions between elements do not happen instantaneously, due to the time required for a signal to propagate, reaction times of individual elements, and so forth. Moreover, time delays are normally nonconstant and may vary with time. This means that it is vital to introduce time delays in any realistic model of neural networks. In order to analyze the fundamental properties of neural networks with time-delayed connections, we consider a system of two coupled two-dimensional nonlinear delay differential equations. This model represents a neural network, where one subsystem receives a delayed input from another subsystem. An exciting feature of the model under consideration is the combination of both discrete and distributed delays, where distributed time delays represent the neural feedback between the two subsystems, and the discrete delays describe the neural interaction within each of the two subsystems. Stability properties are investigated for different commonly used distribution kernels, and the results are compared to the corresponding results on stability for networks with no distributed delays. It is shown how approximations of the boundary of the stability region of a trivial equilibrium can be obtained analytically for the cases of delta, uniform, and weak gamma delay distributions. Numerical techniques are used to investigate stability properties of the fully nonlinear system, and they fully confirm all analytical findings.
Journal of Physiology-paris, 2006
The circuitry of cortical networks involves interacting populations of excitatory (E) and inhibitory (I) neurons whose relationships are now known to a large extent. Inputs to E-and I-cells may have their origins in remote or local cortical areas. We consider a rudimentary model involving E-and I-cells. One of our goals is to test an analytic approach to finding firing rates in neural networks without using a diffusion approximation and to this end we consider in detail networks of excitatory neurons with leaky integrate-and-fire (LIF) dynamics. A simple measure of synchronization, denoted by S q , where q is between 0 and 100 is introduced. Fully connected E-networks have a large tendency to become dominated by synchronously firing groups of cells, except when inputs are relatively weak. We observed random or asynchronous firing in such networks with diverse sets of parameter values. When such firing patterns were found, the analytical approach was often able to accurately predict average neuronal firing rates. We also considered several properties of E-E networks, distinguishing several kinds of firing pattern. Included were those with silences before or after periods of intense activity or with periodic synchronization. We investigated the occurrence of synchronized firing with respect to changes in the internal excitatory postsynaptic potential (EPSP) magnitude in a network of 100 neurons with fixed values of the remaining parameters. When the internal EPSP size was less than a certain value, synchronization was absent. The amount of synchronization then increased slowly as the EPSP amplitude increased until at a particular EPSP size the amount of synchronization abruptly increased, with S 5 attaining the maximum value of 100%. We also found network frequency transfer characteristics for various network sizes and found a linear dependence of firing frequency over wide ranges of the external afferent frequency, with non-linear effects at lower input frequencies. The theory may also be applied to sparsely connected networks, whose firing behaviour was found to change abruptly as the probability of a connection passed through a critical value. The analytical method was also found to be useful for a feed-forward excitatory network and a network of excitatory and inhibitory neurons.
A simple computer model of excitable synaptically connected neurons
1997
Abstract. The space-lumped two-variable neuron model is studied. Extension of the neural model by adding a simple synaptic current allows the demonstration of neural interactions. The production of synchronous burst activity in this simple two-neuron excitatory loop is modeled, including the influence of random background excitatory input. The ability of the neuron model to integrate inputs spatially and temporally is shown. Two refractory periods after stimuli were identified and their role in burst cessation is demonstrated.