Delayed-exponential approximation of a linear homogeneous diffusion model of neuron

The diffusion models of neuronal activity are general yet conceptually simple and flexible enough to be useful in a variety of modeling problems. Unfortunately, even simple diffusion models lead to tedious numerical calculations. Consequently, the existing neural net models use characteristics of a single neuron taken from the pre-diffusion era of neural modeling. Simplistic elements of neural nets forbid to incorporate a single learning neuron structure into the net model. The above drawback cannot be overcome without the use of the adequate structure of the single neuron as an element of a net. A linear (not necessarily homogeneous) diffusion model of a single neuron is a good candidate for such a structure, it must, however, be simplified. In the paper the structure of the diffusion model of neuron is discussed and a linear homogeneous model with reflection is analyzed. For this model an approximation is presented, which is based on the approximation of the first passage time distribution of the Ornstein-Uhlenbeck process by the delayed (shifted) exponential distribution. The resulting model has a simple structure and has a prospective application in neural modeling and in analysis of neural nets.Work supported by Polish Academy of Sciences grant # CPBP 04.01     

A database of computational models of a half-center oscillator for analyzing how neuronal parameters influence network activity

A half-center oscillator (HCO) is a common circuit building block of central pattern generator networks that produce rhythmic motor patterns in animals. Here we constructed an efficient relational database table with the resulting characteristics of the Hill et al.’s (J Comput Neurosci 10:281–302, 2001) HCO simple conductance-based model. The model consists of two reciprocally inhibitory neurons and replicates the electrical activity of the oscillator interneurons of the leech heartbeat central pattern generator under a variety of experimental conditions. Our long-range goal is to understand how this basic circuit building block produces functional activity under a variety of parameter regimes and how different parameter regimes influence stability and modulatability. By using the latest developments in computer technology, we simulated and stored large amounts of data (on the order of terabytes). We systematically explored the parameter space of the HCO and corresponding isolated neuron models using a brute-force approach. We varied a set of selected parameters (maximal conductance of intrinsic and synaptic currents) in all combinations, resulting in about 10 million simulations. We classified these HCO and isolated neuron model simulations by their activity characteristics into identifiable groups and quantified their prevalence. By querying the database, we compared the activity characteristics of the identified groups of our simulated HCO models with those of our simulated isolated neuron models and found that regularly bursting neurons compose only a small minority of functional HCO models; the vast majority was composed of spiking neurons.     

Firing patterns in the adaptive exponential integrate-and-fire model

For simulations of large spiking neuron networks, an accurate, simple and versatile single-neuron modeling framework is required. Here we explore the versatility of a simple two-equation model: the adaptive exponential integrate-and-fire neuron. We show that this model generates multiple firing patterns depending on the choice of parameter values, and present a phase diagram describing the transition from one firing type to another. We give an analytical criterion to distinguish between continuous adaption, initial bursting, regular bursting and two types of tonic spiking. Also, we report that the deterministic model is capable of producing irregular spiking when stimulated with constant current, indicating low-dimensional chaos. Lastly, the simple model is fitted to real experiments of cortical neurons under step current stimulation. The results provide support for the suitability of simple models such as the adaptive exponential integrate-and-fire neuron for large network simulations.     

A modular analog neuron-model for research and teaching

An analog electronic neuron for modeling neuronal networks and for teaching purposes is presented. The circuitry represents a new approach, since ionic channels are all modeled by fast relay switches in combination with resistors. Thus, the synapse design includes nonlinear PSP superposition and reversal potentials. The spike is modeled with Na⁺ and K⁺ currents. Detailed circuit diagrams and examples of performance are given. Besides its use in network research, the model has been successfully used over 3 years of teaching practice.     

Manifestation of function-follow-form in cultured neuronal networks

We expose hidden function-follow-form schemata in the recorded activity of cultured neuronal networks by comparing the activity with simulation results of a new modeling approach. Cultured networks grown from an arbitrary mixture of neuron and glia cells in the absence of external stimulations and chemical cues spontaneously form networks of different sizes (from 50 to several millions of neurons) that exhibit non-arbitrary complex spatio-temporal patterns of activity. The latter is marked by formation of a sequence of synchronized bursting events (SBEs)--short time windows (approximately 200 ms) of rapid neuron firing, separated by longer time intervals (seconds) of sporadic neuron firing. The new dynamical synapse and soma (DSS) model, used here, has been successful in generating sequences of SBEs with the same statistical scaling properties (over six time decades) as those of the small networks. Large networks generate statistically distinct sub-groups of SBEs, each with its own characteristic pattern of neuronal firing ('fingerprint'). This special function (activity) motif has been proposed to emanate from a structural (form) motif--self-organization of the large networks into a fabric of overlapping sub-networks of about 1 mm in size. Here we test this function-follow-form idea by investigating the influence of the connectivity architecture of a model network (form) on the structure of its spontaneous activity (function). We show that a repertoire of possible activity states similar to the observed ones can be generated by networks with proper underlying architecture. For example, networks composed of two overlapping sub-networks exhibit distinct types of SBEs, each with its own characteristic pattern of neuron activity that starts at a specific sub-network. We further show that it is possible to regulate the temporal appearance of the different sub-groups of SBEs by an additional non-synaptic current fed into the soma of the modeled neurons. The ability to regulate the relative temporal ordering of different SBEs might endow the networks with higher plasticity and complexity. These findings call for additional mechanisms yet to be discovered. Recent experimental observations indicate that glia cells coupled to neuronal soma might generate such non-synaptic regulating currents.     

A neuron model as an universal element of self-learning networks for pattern recognition

A mathematical model of the neuron, socalled D-neuron, is proposed on the basis of some new conceptions concerning the molecular mechanism of the synaptical memory. According to these conceptions, the receptors of the neuron reception surface are divided into functional independent fields of receptors. The receptors of any field belong to corresponding membrane protein complex which contains moreover Na⁺-channels, K⁺-channels and eventually other protein subunits. Three processes are supposed to take place in any complex by its interaction with chemical transmitters: i cooperative transitions of the subunits, ii time-controlled transport of ions and iii changes of concentrations of the protein complex subunits. These processes correspond to the following information processings: i recording in the memory, ii discrimination and iii accomodation. In this paper they all are described by an idealized system of algebraic and differential equations. The proposed neuron model can account for the short- and long-term memory mechanism on the level of a single neuron as well as for the control of the neuron networks by the hormones. finally, the neuron model is presented as a universal unit of self-learning networks.     

Information processing in neural networks by means of controlled dynamic regimes

This paper is concerned with the modeling of neural systems regarded as information processing entities. I investigate the various dynamic regimes that are accessible in neural networks considered as nonlinear adaptive dynamic systems. The possibilities of obtaining steady, oscillatory or chaotic regimes are illustrated with different neural network models. Some aspects of the dependence of the dynamic regimes upon the synaptic couplings are examined. I emphasize the role that the various regimes may play to support information processing abilities. I present an example where controlled transient evolutions in a neural network, are used to model the regulation of motor activities by the cerebellar cortex.     

Boolean Models of Genomic Regulatory Networks: Reduction Mappings,Inference, and External Control

Computational modeling of genomic regulation has become an important focus of systems biology and genomic signal processing for the past several years. It holds the promise to uncover both the structure and dynamical properties of the complex gene, protein or metabolic networks responsible for the cell functioning in various contexts and regimes. This, in turn, will lead to the development of optimal intervention strategies for prevention and control of disease. At the same time, constructing such computational models faces several challenges. High complexity is one of the major impediments for the practical applications of the models. Thus, reducing the size/complexity of a model becomes a critical issue in problems such as model selection, construction of tractable subnetwork models, and control of its dynamical behavior. We focus on the reduction problem in the context of two specific models of genomic regulation: Boolean networks with perturbation (BN_P) and probabilistic Boolean networks (PBN). We also compare and draw a parallel between the reduction problem and two other important problems of computational modeling of genomic networks: the problem of network inference and the problem of designing external control policies for intervention/altering the dynamics of the model.     

The linear theory of neuron networks: The static problem

The construction of a theory of activity in neuron networks of arbitrary topological structure is commenced under the linear excitation hypothesis: we consider conditions for possible steady-state equilibria, deferring a dynamical treatment to the sequel.     

Rule-dynamical generalization of McCulloch-Pitts neuron networks

A new aspect for neuronal networks is presented. The aspect is based on the concept of ruledynamics which was originally proposed by one of the authors, Aizawa. The concept of ruledynamics were modeled on the two states cellular automata of neighborhood-three (CA(2/3)). A brief review of ruledynamics is also presented, because most publications of the authors so far have been in Japanese. Our concise assertion in the present paper is that a neuronal network realizes a kind of ruledynamics. This assertion is a speculation on the comparison of McCulloch-Pitts neuron networks with ruledynamics on CA(2/3). A trial is originally shown to demonstrate that a McCulloch-Pitts neuron network can be imitated by an extended version of ruledynamics on CA(2/3).     

Using phase resetting to predict 1:1 and 2:2 locking in two neuron networks in which firing order is not always preserved

Our goal is to understand how nearly synchronous modes arise in heterogenous networks of neurons. In heterogenous networks, instead of exact synchrony, nearly synchronous modes arise, which include both 1:1 and 2:2 phase-locked modes. Existence and stability criteria for 2:2 phase-locked modes in reciprocally coupled two neuron circuits were derived based on the open loop phase resetting curve (PRC) without the assumption of weak coupling. The PRC for each component neuron was generated using the change in synaptic conductance produced by a presynaptic action potential as the perturbation. Separate derivations were required for modes in which the firing order is preserved and for those in which it alternates. Networks composed of two model neurons coupled by reciprocal inhibition were examined to test the predictions. The parameter regimes in which both types of nearly synchronous modes are exhibited were accurately predicted both qualitatively and quantitatively provided that the synaptic time constant is short with respect to the period and that the effect of second order resetting is considered. In contrast, PRC methods based on weak coupling could not predict 2:2 modes and did not predict the 1:1 modes with the level of accuracy achieved by the strong coupling methods. The strong coupling prediction methods provide insight into what manipulations promote near-synchrony in a two neuron network and may also have predictive value for larger networks, which can also manifest changes in firing order. We also identify a novel route by which synchrony is lost in mildly heterogenous networks.     

Neural mechanism of dynamic responses of neurons in inferior temporal cortex in face perception

Understanding the neural mechanisms of object and face recognition is one of the fundamental challenges of visual neuroscience. The neurons in inferior temporal (IT) cortex have been reported to exhibit dynamic responses to face stimuli. However, little is known about how the dynamic properties of IT neurons emerge in the face information processing. To address this issue, we made a model of IT cortex, which performs face perception via an interaction between different IT networks. The model was based on the face information processed by three resolution maps in early visual areas. The network model of IT cortex consists of four kinds of networks, in which the information about a whole face is combined with the information about its face parts and their arrangements. We show here that the learning of face stimuli makes the functional connections between these IT networks, causing a high spike correlation of IT neuron pairs. A dynamic property of subthreshold membrane potential of IT neuron, produced by Hodgkin–Huxley model, enables the coordination of temporal information without changing the firing rate, providing the basis of the mechanism underlying face perception. We show also that the hierarchical processing of face information allows IT cortex to perform a “coarse-to-fine” processing of face information. The results presented here seem to be compatible with experimental data about dynamic properties of IT neurons.     

Spatial firing patterns of auditory neuron network modelling by computer simulation

This communication examines, in digital computer simulated network, input signals and response patterns established at excitatory neurons' level i.e. the membrane potential of neuron soma. It is restricted to spatial patterns of the auditory neuron networks and time factor for nervous conduction and transmission is neglected compared with long maintained membrane potentials of neuron somas. The model analyzes the change in the spatial patterns of the membrane potential in the two dimensional networks of the auditory system. In order to evaluate the contribution of the various parameters, it is started that the simplest model has only one parameter, lateral inhibition. The other parameters are then added, one at a time, to successive models. The lateral inhibition is a necessary condition in the auditory nervous system if any sharpening of the response areas in the single neurons is to occur. A necessary condition for the validity of the model is that it should be applicable to the other senses such as vision and chemical patterns, taste. The threshold feature of auditory neurons aids in producing a sharpening in the neuron of the auditory relay nuclei. It does this clipping the spatial response patterns in one dimensional arrays of excitatory neurons. Recurrent inhibition seems a necessary condition in the sensory nervous system that any kinds of input signals are to be preserved over a wide range of stimulus intensity. In other words, this network has a wide dynamic range against any kinds of input signals. A simple self-recurrent negative feedback does not contribute to the sharpening, but more complex socalled averaged type does. A neuron network is capable of responding stably to stimuli with a wide range of intensity and with any kind of spatial patterns if there is a simple negative feedback mechanism. When there is no negative feedback, input signals soon disappear or saturate in the neuron network. Therefore, recurrent inhibition is the most important mechanism. Spontaneous activity appears to aid in the sharpening by providing a kind of contrast, that is by reducting the amount of activity in neurons adjacent to the excitatory area. Moreover, the effect of spontaneous activity in the model seems to make repples around the excitatory area and suggests that an introduction of activity at any stage of the networks, from whatever source for example reticulum formation and thalamus, might appreciably alter the response patterns at subsequent neuron network. This suggests that the mechanism of the consciousness that might be controlled by the thalamus and or reticular formation. These two dimensional neuron networks may be expanded to three dimensional neuron networks. The former might simulate the auditory nervous system while the latter might simulate the visual system.     

Mesoscopic effects in an agent-based bargaining model in regular lattices

The effect of spatial structure has been proved very relevant in repeated games. In this work we propose an agent based model where a fixed finite population of tagged agents play iteratively the Nash demand game in a regular lattice. The model extends the multiagent bargaining model by Axtell, Epstein and Young modifying the assumption of global interaction. Each agent is endowed with a memory and plays the best reply against the opponent's most frequent demand. We focus our analysis on the transient dynamics of the system, studying by computer simulation the set of states in which the system spends a considerable fraction of the time. The results show that all the possible persistent regimes in the global interaction model can also be observed in this spatial version. We also find that the mesoscopic properties of the interaction networks that the spatial distribution induces in the model have a significant impact on the diffusion of strategies, and can lead to new persistent regimes different from those found in previous research. In particular, community structure in the intratype interaction networks may cause that communities reach different persistent regimes as a consequence of the hindering diffusion effect of fluctuating agents at their borders.     

First return maps for the dynamics of synaptically coupled conditional bursters

The pre-Bötzinger complex (preBötc) in the mammalian brainstem has an important role in generating respiratory rhythms. An influential differential equation model for the activity of individual neurons in the preBötc yields transitions from quiescence to bursting to tonic spiking as a parameter is varied. Further, past work has established that bursting dynamics can arise from a pair of tonic model cells coupled with synaptic excitation. In this paper, we analytically derive one- and two-dimensional maps from the differential equations for a self-coupled neuron and a two-neuron network, respectively. Using a combination of analysis and simulations of these maps, we explore the possible forms of dynamics that the model networks can produce as well as which transitions between dynamic regimes are mathematically possible.     

Interactions of persistent sodium and calcium-activated nonspecific cationic currents yield dynamically distinct bursting regimes in a model of respiratory neurons

The preBötzinger complex (preBötC) is a heterogeneous neuronal network within the mammalian brainstem that has been experimentally found to generate robust, synchronous bursts that drive the inspiratory phase of the respiratory rhythm. The persistent sodium (NaP) current is observed in every preBötC neuron, and significant modeling effort has characterized its contribution to square-wave bursting in the preBötC. Recent experimental work demonstrated that neurons within the preBötC are endowed with a calcium-activated nonspecific cationic (CAN) current that is activated by a signaling cascade initiated by glutamate. In a preBötC model, the CAN current was shown to promote robust bursts that experience depolarization block (DB bursts). We consider a self-coupled model neuron, which we represent as a single compartment based on our experimental finding of electrotonic compactness, under variation of g _NaP, the conductance of the NaP current, and g _CAN, the conductance of the CAN current. Varying these two conductances yields a spectrum of activity patterns, including quiescence, tonic activity, square-wave bursting, DB bursting, and a novel mixture of square-wave and DB bursts, which match well with activity that we observe in experimental preparations. We elucidate the mechanisms underlying these dynamics, as well as the transitions between these regimes and the occurrence of bistability, by applying the mathematical tools of bifurcation analysis and slow-fast decomposition. Based on the prevalence of NaP and CAN currents, we expect that the generalizable framework for modeling their interactions that we present may be relevant to the rhythmicity of other brain areas beyond the preBötC as well.     

A Population Density Approach That Facilitates Large-Scale Modeling of Neural Networks: Analysis and an Application to Orientation Tuning

We explore a computationally efficient method of simulating realistic networks of neurons introduced by Knight, Manin, and Sirovich (1996) in which integrate-and-fire neurons are grouped into large populations of similar neurons. For each population, we form a probability density that represents the distribution of neurons over all possible states. The populations are coupled via stochastic synapses in which the conductance of a neuron is modulated according to the firing rates of its presynaptic populations. The evolution equation for each of these probability densities is a partial differential-integral equation, which we solve numerically. Results obtained for several example networks are tested against conventional computations for groups of individual neurons.We apply this approach to modeling orientation tuning in the visual cortex. Our population density model is based on the recurrent feedback model of a hypercolumn in cat visual cortex of Somers et al. (1995). We simulate the response to oriented flashed bars. As in the Somers model, a weak orientation bias provided by feed-forward lateral geniculate input is transformed by intracortical circuitry into sharper orientation tuning that is independent of stimulus contrast.The population density approach appears to be a viable method for simulating large neural networks. Its computational efficiency overcomes some of the restrictions imposed by computation time in individual neuron simulations, allowing one to build more complex networks and to explore parameter space more easily. The method produces smooth rate functions with one pass of the stimulus and does not require signal averaging. At the same time, this model captures the dynamics of single-neuron activity that are missed in simple firing-rate models.     

Modeling epidemics dynamics on heterogenous networks

The dynamics of the SIS process on heterogenous networks, where different local communities are connected by airlines, is studied. We suggest a new modeling technique for travelers movement, in which the movement does not affect the demographic parameters characterizing the metapopulation. A solution to the deterministic reaction-diffusion equations that emerges from this model on a general network is presented. A typical example of a heterogenous network, the star structure, is studied in detail both analytically and using agent-based simulations. The interplay between demographic stochasticity, spatial heterogeneity and the infection dynamics is shown to produce some counterintuitive effects. In particular it was found that, while movement always increases the chance of an outbreak, it may decrease the steady-state fraction of sick individuals. The importance of the modeling technique in estimating the outcomes of a vaccination campaign is demonstrated.     

Competition between Homophily and Information Entropy Maximization in Social Networks

In social networks, it is conventionally thought that two individuals with more overlapped friends tend to establish a new friendship, which could be stated as homophily breeding new connections. While the recent hypothesis of maximum information entropy is presented as the possible origin of effective navigation in small-world networks. We find there exists a competition between information entropy maximization and homophily in local structure through both theoretical and experimental analysis. This competition suggests that a newly built relationship between two individuals with more common friends would lead to less information entropy gain for them. We demonstrate that in the evolution of the social network, both of the two assumptions coexist. The rule of maximum information entropy produces weak ties in the network, while the law of homophily makes the network highly clustered locally and the individuals would obtain strong and trust ties. A toy model is also presented to demonstrate the competition and evaluate the roles of different rules in the evolution of real networks. Our findings could shed light on the social network modeling from a new perspective.