Density dependent neurodynamics

The dynamics of a neural network depends on density parameters at (at least) two different levels: the subcellular density of ion channels in single neurons, and the density of cells and synapses at a network level. For the Frankenhaeuser-Huxley (FH) neural model, the density of sodium (Na) and potassium (K) channels determines the behaviour of a single neuron when exposed to an external stimulus. The features of the onset of single neuron oscillations vary qualitatively among different regions in the channel density plane. At a network level, the density of neurons is reflected in the global connectivity. We study the relation between the two density levels in a network of oscillatory FH neurons, by qualitatively distinguishing between three regions, where the mean network activity is (1) spiking, (2) oscillating with enveloped frequencies, and (3) bursting, respectively. We demonstrate that the global activity can be shifted between regions by changing either the density of ion channels at the subcellular level, or the connectivity at the network level, suggesting that different underlying mechanisms can explain similar global phenomena. Finally, we model a possible effect of anaesthesia by blocking specific inhibitory ion channels.     

Moment fitting for parameter inference in repeatedly and partially observed stochastic biological models

The inference of reaction rate parameters in biochemical network models from time series concentration data is a central task in computational systems biology. Under the assumption of well mixed conditions the network dynamics are typically described by the chemical master equation, the Fokker Planck equation, the linear noise approximation or the macroscopic rate equation. The inverse problem of estimating the parameters of the underlying network model can be approached in deterministic and stochastic ways, and available methods often compare individual or mean concentration traces obtained from experiments with theoretical model predictions when maximizing likelihoods, minimizing regularized least squares functionals, approximating posterior distributions or sequentially processing the data. In this article we assume that the biological reaction network can be observed at least partially and repeatedly over time such that sample moments of species molecule numbers for various time points can be calculated from the data. Based on the chemical master equation we furthermore derive closed systems of parameter dependent nonlinear ordinary differential equations that predict the time evolution of the statistical moments. For inferring the reaction rate parameters we suggest to not only compare the sample mean with the theoretical mean prediction but also to take the residual of higher order moments explicitly into account. Cost functions that involve residuals of higher order moments may form landscapes in the parameter space that have more pronounced curvatures at the minimizer and hence may weaken or even overcome parameter sloppiness and uncertainty. As a consequence both deterministic and stochastic parameter inference algorithms may be improved with respect to accuracy and efficiency. We demonstrate the potential of moment fitting for parameter inference by means of illustrative stochastic biological models from the literature and address topics for future research.     

A novel connectionist framework for computation of an approximate convex-hull of a set of planar points, circles and ellipses

We propose a two layer neural network for computation of an approximate convex-hull of a set of points or a set of circles/ellipses of different sizes. The algorithm is based on a very elegant concept - shrinking of a rubber band surrounding the set of planar objects. Logically, a set of neurons is placed on a circle (rubber band) surrounding the objects. Each neuron has a parameter vector associated with it. This may be viewed as the current position of the neuron. The given set of points/objects exerts a force of attraction on every neuron, which determines how its current position will be updated (as if, the force determines the direction of movement of the neuron lying on the rubber band). As the network evolves, the neurons (parameter vectors) approximate the convex-hull more and more accurately. The scheme can be applied to find the convex-hull of a planar set of circles or ellipses or a mixture of the two. Some properties related to the evolution of the algorithm are also presented.     

Exploring how extracellular electric field modulates neuron activity through dynamical analysis of a two-compartment neuron model

To investigate how extracellular electric field modulates neuron activity, a reduced two-compartment neuron model in the presence of electric field is introduced in this study. Depending on neuronal geometric and internal coupling parameters, the behaviors of the model have been studied extensively. The neuron model can exist in quiescent state or repetitive spiking state in response to electric field stimulus. Negative electric field mainly acts as inhibitory stimulus to the neuron, positive weak electric field could modulate spiking frequency and spike timing when the neuron is already active, and positive electric fields with sufficient intensity could directly trigger neuronal spiking in the absence of other stimulations. By bifurcation analysis, it is observed that there is saddle-node on invariant circle bifurcation, supercritical Hopf bifurcation and subcritical Hopf bifurcation appearing in the obtained two parameter bifurcation diagrams. The bifurcation structures and electric field thresholds for triggering neuron firing are determined by neuronal geometric and coupling parameters. The model predicts that the neurons with a nonsymmetric morphology between soma and dendrite, are more sensitive to electric field stimulus than those with the spherical structure. These findings suggest that neuronal geometric features play a crucial role in electric field effects on the polarization of neuronal compartments. Moreover, by determining the electric field threshold of our biophysical model, we could accurately distinguish between suprathreshold and subthreshold electric fields. Our study highlights the effects of extracellular electric field on neuronal activity from the biophysical modeling point of view. These insights into the dynamical mechanism of electric field may contribute to the investigation and development of electromagnetic therapies, and the model in our study could be further extended to a neuronal network in which the effects of electric fields on network activity may be investigated.     

The response of cortical neurons to in vivo-like input current: theory and experiment: II. Time-varying and spatially distributed inputs

The response of a population of neurons to time-varying synaptic inputs can show a rich phenomenology, hardly predictable from the dynamical properties of the membrane’s inherent time constants. For example, a network of neurons in a state of spontaneous activity can respond significantly more rapidly than each single neuron taken individually. Under the assumption that the statistics of the synaptic input is the same for a population of similarly behaving neurons (mean field approximation), it is possible to greatly simplify the study of neural circuits, both in the case in which the statistics of the input are stationary (reviewed in La Camera et al. in Biol Cybern, 2008) and in the case in which they are time varying and unevenly distributed over the dendritic tree. Here, we review theoretical and experimental results on the single-neuron properties that are relevant for the dynamical collective behavior of a population of neurons. We focus on the response of integrate-and-fire neurons and real cortical neurons to long-lasting, noisy, in vivo-like stationary inputs and show how the theory can predict the observed rhythmic activity of cultures of neurons. We then show how cortical neurons adapt on multiple time scales in response to input with stationary statistics in vitro. Next, we review how it is possible to study the general response properties of a neural circuit to time-varying inputs by estimating the response of single neurons to noisy sinusoidal currents. Finally, we address the dendrite–soma interactions in cortical neurons leading to gain modulation and spike bursts, and show how these effects can be captured by a two-compartment integrate-and-fire neuron. Most of the experimental results reviewed in this article have been successfully reproduced by simple integrate-and-fire model neurons.     

Spike independency in feed-forward networks

A cortical neuron puts thousands of synaptic contacts on other neurons. The effect of the spike event spreads over a large number of neurons. So it is possible for spike timings to be correlated to each other. But there have not been so many reports of spike timing correlations, while there have been many reports of somewhat longer time range correlations through mean spike rates. Can independent firings be preserved in spite of a number of connections? The present study attempts to determine whether independent firings can be propagated through a simple feed-forward neural network. It is assumed that each unit obeys a threshold mechanism at each discrete time and that connections are statistically uniform with the excitation balanced to the inhibition and delay distributed. It is found that the independent firings can be stably propagated through the feed-forward network at a network parameter region, which contains the physiologically reasonable range. Another interesting result is that the independency-stable spike probability has a lower limit 0.0323.     

Weak electric fields detectability in a noisy neural network

We investigate the detectability of weak electric field in a noisy neural network based on Izhikevich neuron model systematically. The neural network is composed of excitatory and inhibitory neurons with similar ratio as that in the mammalian neocortex, and the axonal conduction delays between neurons are also considered. It is found that the noise intensity can modulate the detectability of weak electric field. Stochastic resonance (SR) phenomenon induced by white noise is observed when the weak electric field is added to the network. It is interesting that SR almost disappeared when the connections between neurons are cancelled, suggesting the amplification effects of the neural coupling on the synchronization of neuronal spiking. Furthermore, the network parameters, such as the connection probability, the synaptic coupling strength, the scale of neuron population and the neuron heterogeneity, can also affect the detectability of the weak electric field. Finally, the model sensitivity is studied in detail, and results show that the neural network model has an optimal region for the detectability of weak electric field signal.     

Generating oscillatory bursts from a network of regular spiking neurons without inhibition

Avian nucleus isthmi pars parvocellularis (Ipc) neurons are reciprocally connected with the layer 10 (L10) neurons in the optic tectum and respond with oscillatory bursts to visual stimulation. Our in vitro experiments show that both neuron types respond with regular spiking to somatic current injection and that the feedforward and feedback synaptic connections are excitatory, but of different strength and time course. To elucidate mechanisms of oscillatory bursting in this network of regularly spiking neurons, we investigated an experimentally constrained model of coupled leaky integrate-and-fire neurons with spike-rate adaptation. The model reproduces the observed Ipc oscillatory bursting in response to simulated visual stimulation. A scan through the model parameter volume reveals that Ipc oscillatory burst generation can be caused by strong and brief feedforward synaptic conductance changes. The mechanism is sensitive to the parameter values of spike-rate adaptation. In conclusion, we show that a network of regular-spiking neurons with feedforward excitation and spike-rate adaptation can generate oscillatory bursting in response to a constant input.     

A burst-feedback model of fast-phase burst generation during nystagmus

 Vestibular and optokinetic nystagmus are characterized by alternating slow-phase eye rotations that stabilize the retinal image, and fast-phase eye rotations that reset eye position. Nystagmus is coordinated in the brainstem by burst neurons that fire an intense, temporally circumscribed burst that terminates the slow phase and drives the fast phase. This paper demonstrates that such a burst can be generated during nystagmus using a simple neural network model containing only known brainstem neurons and their interconnections. These include the feedback connections of the burst neuron (burst feedback). The burst neuron excites itself directly, and disinhibits itself by inhibiting the pause neuron (positive feedback). It also inhibits itself by inhibiting the vestibular neuron (negative feedback). The burst neuron begins to fire after its inhibitory bias is overcome by excitation from the vestibular neuron, and burst neuron positive feedback then produces an intense burst with an abrupt onset. The burst causes the vestibular and pause neurons to pause. The benefit of the pause neuron loop is that it contributes to burst neuron positive feedback when it is needed at burst onset, but that contribution is eliminated when the pause neuron pauses and opens the loop. The burst can then terminate, with an offset duration proportional to burst amplitude, under the control of burst neuron self-excitation and inhibitory bias. Model neuron behavior is comparable to that of real brainstem neurons. Synchronized bursts can be produced over the population of burst neurons in a distributed version of the network. Variability in connection weights in the distributed network results in variability in prelude activity among burst neurons that is similar to the spread in lead observed for real burst neurons during nystagmus. Received: 11 April 1996 / Accepted in revised form: 6 August 1996     

A Many-Body Field Theory Approach to Stochastic Models in Population Biology

Background

Many models used in theoretical ecology, or mathematical epidemiology are stochastic, and may also be spatially-explicit. Techniques from quantum field theory have been used before in reaction-diffusion systems, principally to investigate their critical behavior. Here we argue that they make many calculations easier and are a possible starting point for new approximations.

Methodology

We review the many-body field formalism for Markov processes and illustrate how to apply it to a ‘Brownian bug’ population model, and to an epidemic model. We show how the master equation and the moment hierarchy can both be written in particularly compact forms. The introduction of functional methods allows the systematic computation of the effective action, which gives the dynamics of mean quantities. We obtain the 1-loop approximation to the effective action for general (space-) translation invariant systems, and thus approximations to the non-equilibrium dynamics of the mean fields.

Conclusions

The master equations for spatial stochastic systems normally take a neater form in the many-body field formalism. One can write down the dynamics for generating functional of physically-relevant moments, equivalent to the whole moment hierarchy. The 1-loop dynamics of the mean fields are the same as those of a particular moment-closure.     

Effects of host social hierarchy on disease persistence

The effects of social hierarchy on population dynamics and epidemiology are examined through a model which contains a number of fundamental features of hierarchical systems, but is simple enough to allow analytical insight. In order to allow for differences in birth rates, contact rates and movement rates among different sets of individuals the population is first divided into subgroups representing levels in the hierarchy. Movement, representing dominance challenges, is allowed between any two levels, giving a completely connected network. The model includes hierarchical effects by introducing a set of dominance parameters which affect birth rates in each social level and movement rates between social levels, dependent upon their rank. Although natural hierarchies vary greatly in form, the skewing of contact patterns, introduced here through non-uniform dominance parameters, has marked effects on the spread of disease. A simple homogeneous mixing differential equation model of a disease with SI dynamics in a population subject to simple birth and death process is presented and it is shown that the hierarchical model tends to this as certain parameter regions are approached. Outside of these parameter regions correlations within the system give rise to deviations from the simple theory. A Gaussian moment closure scheme is developed which extends the homogeneous model in order to take account of correlations arising from the hierarchical structure, and it is shown that the results are in reasonable agreement with simulations across a range of parameters. This approach helps to elucidate the origin of hierarchical effects and shows that it may be straightforward to relate the correlations in the model to measurable quantities which could be used to determine the importance of hierarchical corrections. Overall, hierarchical effects decrease the levels of disease present in a given population compared to a homogeneous unstructured model, but show higher levels of disease than structured models with no hierarchy. The separation between these three models is greatest when the rate of dominance challenges is low, reducing mixing, and when the disease prevalence is low. This suggests that these effects will often need to be considered in models being used to examine the impact of control strategies where the low disease prevalence behaviour of a model is critical.     

The response of cortical neurons to in vivo-like input current: theory and experiment

The study of several aspects of the collective dynamics of interacting neurons can be highly simplified if one assumes that the statistics of the synaptic input is the same for a large population of similarly behaving neurons (mean field approach). In particular, under such an assumption, it is possible to determine and study all the equilibrium points of the network dynamics when the neuronal response to noisy, in vivo-like, synaptic currents is known. The response function can be computed analytically for simple integrate-and-fire neuron models and it can be measured directly in experiments in vitro. Here we review theoretical and experimental results about the neural response to noisy inputs with stationary statistics. These response functions are important to characterize the collective neural dynamics that are proposed to be the neural substrate of working memory, decision making and other cognitive functions. Applications to the case of time-varying inputs are reviewed in a companion paper (Giugliano et al. in Biol Cybern, 2008). We conclude that modified integrate-and-fire neuron models are good enough to reproduce faithfully many of the relevant dynamical aspects of the neuronal response measured in experiments on real neurons in vitro.     

Inhibitory synchrony as a mechanism for attentional gain modulation

Recordings from area V4 of monkeys have revealed that when the focus of attention is on a visual stimulus within the receptive field of a cortical neuron, two distinct changes can occur: The firing rate of the neuron can change and there can be an increase in the coherence between spikes and the local field potential (LFP) in the gamma-frequency range (30-50 Hz). The hypothesis explored here is that these observed effects of attention could be a consequence of changes in the synchrony of local interneuron networks. We performed computer simulations of a Hodgkin-Huxley type neuron driven by a constant depolarizing current, I, representing visual stimulation and a modulatory inhibitory input representing the effects of attention via local interneuron networks. We observed that the neuron's firing rate and the coherence of its output spike train with the synaptic inputs was modulated by the degree of synchrony of the inhibitory inputs. When inhibitory synchrony increased, the coherence of spiking model neurons with the synaptic input increased, but the firing rate either increased or remained the same. The mean number of synchronous inhibitory inputs was a key determinant of the shape of the firing rate versus current (f-I) curves. For a large number of inhibitory inputs (approximately 50), the f-I curve saturated for large I and an increase in input synchrony resulted in a shift of sensitivity-the model neuron responded to weaker inputs I. For a small number (approximately 10), the f-I curves were non-saturating and an increase in input synchrony led to an increase in the gain of the response-the firing rate in response to the same input was multiplied by an approximately constant factor. The firing rate modulation with inhibitory synchrony was highest when the input network oscillated in the gamma frequency range. Thus, the observed changes in firing rate and coherence of neurons in the visual cortex could be controlled by top-down inputs that regulated the coherence in the activity of a local inhibitory network discharging at gamma frequencies.     

An extended model for a spiking neuron class

This paper proposes an extension to the model of a spiking neuron for information processing in artificial neural networks, developing a new approach for the dynamic threshold of the integrate-and-fire neuron. This new approach invokes characteristics of biological neurons such as the behavior of chemical synapses and the receptor field. We demonstrate how such a digital model of spiking neurons can solve complex nonlinear classification with a single neuron, performing experiments for the classical XOR problem. Compared with rate-coded networks and the classical integrate-and-fire model, the trained network demonstrated faster information processing, requiring fewer neurons and shorter learning periods. The extended model validates all the logic functions of biological neurons when such functions are necessary for the proper flow of binary codes through a neural network.     

Vibrational resonance in a randomly connected neural network

A randomly connected network is constructed with similar characteristics (e.g., the ratio of excitatory and inhibitory neurons, the connection probability between neurons, and the axonal conduction delays) as that in the mammalian neocortex and the effects of high-frequency electrical field on the response of the network to a subthreshold low-frequency electrical field are studied in detail. It is found that both the amplitude and frequency of the high-frequency electrical field can modulate the response of the network to the low-frequency electric field. Moreover, vibrational resonance (VR) phenomenon induced by the two types of electrical fields can also be influenced by the network parameters, such as the neuron population, the connection probability between neurons and the synaptic strength. It is interesting that VR is found to be related with the ratio of excitatory neurons that are under high-frequency electrical stimuli. In summary, it is suggested that the interaction of excitatory and inhibitory currents is also an important factor that can influence the performance of VR in neural networks.     

Analysis of stability of neural network with inhibitory neurons

Phase coding in a neural network composed of neural oscillators with inhibitory neurons was studied based on the theory of stochastic phase dynamics. We found that with increasing the coupling coefficients of inhibitory neural oscillators, the firing density in excitatory population transits to a critical state. In this case, when we increase the inhibitory coupling, the firing density will come into dynamic balance again and tend to a fixed value gradually. According to the phenomenon, in the paper we found parameter regions to exhibit those different population states, called dividing zones including flat fading zone, rapid fading zone and critical zone. Based on the dividing zones we can choose the number ratio between inhibitory neurons and excitatory neurons in the neural network, and estimate the coupling action of inhibitory population and excitatory population. Our research also shows that the balance value, enabling the firing density to reach the dynamic balance, does not depend on initial conditions. In addition, the critical value in critical state is only related to the number ratio between inhibitory neurons and excitatory neurons, but is independent of inhibitory coupling and excitatory coupling.     

A general mathematical framework for the analysis of spatiotemporal point processes

Spatial and stochastic models are often straightforward to simulate but difficult to analyze mathematically. Most of the mathematical methods available for nonlinear stochastic and spatial models are based on heuristic rather than mathematically justified assumptions, so that, e.g., the choice of the moment closure can be considered more of an art than a science. In this paper, we build on recent developments in specific branch of probability theory, Markov evolutions in the space of locally finite configurations, to develop a mathematically rigorous and practical framework that we expect to be widely applicable for theoretical ecology. In particular, we show how spatial moment equations of all orders can be systematically derived from the underlying individual-based assumptions. Further, as a new mathematical development, we go beyond mean-field theory by discussing how spatial moment equations can be perturbatively expanded around the mean-field model. While we have suggested such a perturbation expansion in our previous research, the present paper gives a rigorous mathematical justification. In addition to bringing mathematical rigor, the application of the mathematically well-established framework of Markov evolutions allows one to derive perturbation expansions in a transparent and systematic manner, which we hope will facilitate the application of the methods in theoretical ecology.     

Synaptic Plasticity Enables Adaptive Self-Tuning Critical Networks

During rest, the mammalian cortex displays spontaneous neural activity. Spiking of single neurons during rest has been described as irregular and asynchronous. In contrast, recent in vivo and in vitro population measures of spontaneous activity, using the LFP, EEG, MEG or fMRI suggest that the default state of the cortex is critical, manifested by spontaneous, scale-invariant, cascades of activity known as neuronal avalanches. Criticality keeps a network poised for optimal information processing, but this view seems to be difficult to reconcile with apparently irregular single neuron spiking. Here, we simulate a 10,000 neuron, deterministic, plastic network of spiking neurons. We show that a combination of short- and long-term synaptic plasticity enables these networks to exhibit criticality in the face of intrinsic, i.e. self-sustained, asynchronous spiking. Brief external perturbations lead to adaptive, long-term modification of intrinsic network connectivity through long-term excitatory plasticity, whereas long-term inhibitory plasticity enables rapid self-tuning of the network back to a critical state. The critical state is characterized by a branching parameter oscillating around unity, a critical exponent close to -3/2 and a long tail distribution of a self-similarity parameter between 0.5 and 1.     

Towards an artificial brain

Three components of a brain model operating on neuromolecular computing principles are described. The first component comprises neurons whose input-output behavior is controlled by significant internal dynamics. Models of discrete enzymatic neurons, reaction-diffusion neurons operating on the basis of the cyclic nucleotide cascade, and neurons controlled by cytoskeletal dynamics are described. The second component of the model is an evolutionary learning algorithm which is used to mold the behavior of enzyme-driven neurons or small networks of these neurons for specific function, usually pattern recognition or target seeking tasks. The evolutionary learning algorithm may be interpreted either as representing the mechanism of variation and natural selection acting on a phylogenetic time scale, or as a conceivable ontogenetic adaptation mechanism. The third component of the model is a memory manipulation scheme, called the reference neuron scheme. In principle it is capable of orchestrating a repertoire of enzyme-driven neurons for coherent function. The existing implementations, however, utilize simple neurons without internal dynamics. Spatial navigation and simple game playing (using tic-tac-toe) provide the task environments that have been used to study the properties of the reference neuron model. A memory-based evolutionary learning algorithm has been developed that can assign credit to the individual neurons in a network. It has been run on standard benchmark tasks, and appears to be quite effective both for conventional neural nets and for networks of discrete enzymatic neurons. The models have the character of artificial worlds in that they map the hierarchy of processes in the brain (at the molecular, neuronal, and network levels), provide a task environment, and use this relatively self-contained setup to develop and evaluate learning and adaptation algorithms.