Irregular persistent activity in a model of a local cortical network
Francesca BARBIERI (Hospital Clínic - IDIBAPS)

Abstract: Recent neuro-physiological experiments on monkeys (Compte et al. 2003) have reported highly irregular persistent activity during the performance of an oculomotor delayed-response task. These experiments show that during the delay period the ISI's coefficient of variation (CV) of prefrontal neurons is above 1, on average, and larger than during the fixation period, regardless of whether the cue is preferred or nonpreferred. Previous models (Amit and Brunel 1997, Brunel 2000) of spontaneous and selective persistent activity in the cortex based on excitatory synaptic feedback do not reproduce this feature because the excitatory feedback during persistent activity brings neurons in a region of the f-I curve in which the firing is relatively independent from fluctuations and hence the CV is small. To overcome this problem, we introduced two ingredients: (1) a high post-spike reset potential (close to threshold) , (2) a non-linear relationship between synaptic efficacy and pre-synaptic firing rate via a short-term depression (STD) mechanism. We show that when the reset potential is close enough to the threshold, the CV-I curve has a maximum above 1 for a sub-threshold mean current. The range of the mean synaptic input values for which the CV is greater than 1 is always in the sub-threshold regime in which firing is dominated by fluctuations of the mean synaptic input. With short-term depression, synaptic efficacies saturate at a certain limiting value of the presynaptic frequency; this is turn provokes a saturation of the mean synaptic current to a neuron at the same limiting presynaptic frequency. This allows the persistent state solution to reach the region of the f-I curve which corresponds to high values of the CV. We tested this idea both with numerical simulations and analytical techniques. For the analytical studies we used mean- field techniques, recently extended in presence of STD (Romani et al. 2006), that involves the use of the distribution of the interspikes intervals of an integrate-and-fire neuron receiving a Gaussian current in input; this permits to obtain an accurate estimate of the statistic of the postsynaptic current in presence of STD and hence to find the stationary states in a self-consistent way. We simulated a fully connected excitatory network of leaky integrate-and-fire neurons endowed with STD and we found a very good agreement with theoretical prediction for a large range of synaptic efficacies.

Stochastic models of excitable systems
Nils BERGLUND, (Université d'Orléans)

Abstract: The generation of action potentials in neurons is sometimes described by differential equations with two well-separated time scale. We shall describe the effect of noise on such equations, in particular on the statistics of interspike times. (Joint work with Barbara Gentz, University of Bielefeld.)

Fokker-Planck models in neuroscience
María José CÁCERES (Univesidad de Granada)

Abstract: We are interested in a Fokker-Planck model, which describes the behavior of neuronal networks in the mammalian neocortex. In the mammalian brains there are around $10^{10}$ neurons, which represents a sufficiently large number to think of a kinetic approach as appropriate. In [1], the authors presented a detailed theorical framework for statistical descriptions of neuronal networks and derived 2-dimensional kinetic equations directly from conductance-based integrate-and-fire neuronal networks (see [2]). They reduced the dimension via moment closures and also described different limits of these kinetic equations. In our present work we are studying a numerical method to simulate the Fokker-Planck equations developed for these models. We try to numerically validate the moment clousures and limits presented in [1], comparing numerically the stationary solutions of these models with the stationary distribution function of the Fokker-Planck equations. Our solver allows us to obtain the numerical evolution of the solution and to consider the distribution function depending of all the variables.

Physics of viral entry
Tom CHOU (Univesity of California at Los Angeles)

Abstract: I will develop and analyze three stochastic models used to describe how certain viruses enter the cell. The first model concerns the stochastic engagement of receptors and the entry mechanisms of viruses. The second model describes the transport of virus material to to the host cell nucleus, while the third model is a revisit of the classic translocation problem used to model how polymers enter through membrane pores.

Nested oscillations in the emergent activity of cortical networks
Albert COMPTE (Hospital Clínic - IDIBAPS)

Abstract: Neural activity in the cerebral cortex is characterized by temporal dynamics at a variety of temporal frequencies. Two frequency bands have been shown to rely primarily on mechanisms of the local cortical circuit: the slow oscillation band (< 1 Hz) and the beta/gamma-frequency band (20-80 Hz). I will present simulation results from a computational model of the cortical microcircuit showing the generation of these nested oscillations, in agreement with the experimental data obtained in cortical slices in vitro. The model consists of 1,000 excitatory neurons and 250 inhibitory neurons modeled according to the Hodgkin-Huxley formalism, including physiologically identified ion channels of cortical neurons, and interconnected through conductance-based synapses. The model identifies key physiological mechanisms that underlie the generation of the various rhythms and their propagation across the network.

Dynamics of Morris-Lecar models
Stephen COOMBES (Univesity of Nottingham)

Abstract: The Morris-Lecar (ML) neuron model is a two dimensional conductance based model that is often used as an idealised fast-spiking pyramidal neuron. Its planar nature has encouraged much analysis of the single neuron model using tools from phase-plane analysis and the "geometry of excitability". When treating synaptic or gap junction coupled networks of oscillating ML neurons these techniques are the natural basis for developing a weakly-coupled oscillator theory. However, to probe network dynamics in the strong coupling regime requires an alternative approach. I will show how results in this area can be obtained by using a piece-wise linear caricature of the ML model. In illustration of the usefulness of such an approach I will first consider gap junction coupling and show how to analyse emergent fluctuations in the mean membrane potential (as instabilities of an asynchronous network state). Next I will treat synaptically coupled networks with a phenomenological form of retrograde second messenger signalling that can support depolarisation induced suppression of excitation. In this case I will describe a mechanism for the emergence of ultra-low frequency synchronised oscillations.

Numerical computationn of slow manifolds and canard orbits near a folded node - Application to neuronal systems
Mathieu DESROCHES (Bristol University)

Abstract: We investigate the geometry of two-dimensional slow manifolds and the organisation of associated canard solutions in three-dimensional slow-fast vector fields with two slow variables. To this end, we introduce a numerical method, based on the numerical solution of families of two-point boundary value problems using continuation techniques, that allows to compute the slow manifolds. We can then identify and compute canard orbits as individual objects and follow them in parameter space, that is, we compute one-parameter families of canard solutions in dependence of a given system's parameter. In this way we get new insight into their underlying bifurcation structure. We will present this numerical method in the case of two neuronal systems, namely, the self-coupled FitzHugh-Nagumo system and a three-dimensional reduction of the Hodgkin-Huxley equation. We show how canard orbits organise the geometry and the dynamics of associated mixed-mode oscillations.

Patterns, stability and collapse for two-dimensional biological swarms
Maria Rita d'ORSOGNA (California State University at Northridge)

Abstract: Schools of fish, flocks of birds and swarms of insects self-organize in response to external stimuli or by direct interaction, forming beautiful, coherent patterns. How do these arise? What are their properties? How are individual characteristics linked to collective behaviours? In this talk we discuss the swarming of a non-linear system of self propelled agents that interact via pairwise attractive and repulsive potentials. We are able to predict distinct aggregation morphologies, such as flocks and vortices, by utilizing statistical mechanics tools. We also relate the interaction potential to the collapsing or dispersing behaviour of agents. A kinetic theory is derived and we analyze the continuum limit of our model. Finally, we also discuss possible applications to the development of artificial swarming teams.

United by noise: randomness helps swarms stay together
Carlos ESCUDERO (Instituto de Matemáticas y Física Fundamental - CSIC)

Abstract: Amongst the most striking aspects of the movement of many animal groups are their sudden coherent changes in direction. Recent observations of locusts and starlings have shown that this directional switching is an intrinsic property of their motion. Similar direction switches are seen in self-propelled particle (SPP) and other models of group motion. Comprehending the factors which determine such switches is key to understanding the movement of these groups. Here we adopt a coarse-grained approach to the study of directional switching in a SPP model assuming an underlying one-dimensional Fokker-Planck equation (FPE) for the mean velocity of the particles. We continue with this assumption in analysing experimental data on locusts and use a similar systematic FPE coefficient estimation approach to extract the relevant information for the assumed FPE underlying that data. We determine the mean time between direction switches as a function of group density for the SPP model. This systematic approach allows us to identify key differences between the SPP model and the data, revealing that individual locusts increase the randomness of their movements in response to a loss of alignment by the group. We give a quantitative description of how locusts use noise to maintain swarm alignment. We discuss further how properties of individual animal behaviour, inferred using the FPE coefficient estimation approach, can be implemented in the SPP model in order to replicate qualitatively the group level dynamics seen in the experimental data.

Mean-field descriptions of populations of neurons
Olivier FAUGERAS (Univesité Sophie-Antipolis)

Abstract: We present a general approach to the analysis of large sets of neurons. It is firmly grounded in the mathematics of stochastic analysis and principles of large deviations. We start from a description of individual neurons as obeying a set of stochastic differential equations (SDEs) such as those of Hodgkin-Huxley, a description of their synaptic interactions, and proceed to develop a mean-field analysis and description of the activity of each neuronal population when the number of neurons in each population becomes very large. Unlike other approaches, we rigorously establish the SDESs that describe each individual population. These SDEs turn out to be non-Markovian and display a strong coupling between the moments of the probabilistic laws describing the population membrane potentials. We relate them to some popular neural-mass equations such as those established by Jansen and Rit that are quite heavily used in the modeling of such signals as EEG and MEG. In doing so we discover that the coupling between the mean membrane potentials and their covariances that results from our equations significantly alters the time response of the populations to a given stimulus. This fact seems to have been overlooked in previous work. We discuss some of the consequences of our analysis on the modeling of neural populations.

Excitable dynamics of cell regulation
Jordi GARCÍA-OJALVO (Univesitat Politècnica de Catalunya)

Abstract: Excitability is a paradigmatic dynamical state that has been found to have important implications in biology, especially in the field of neuronal dynamics. In this talk I will discuss examples of excitable dynamics in a different type of biological process, namely gene regulation. In particular, I will first consider the competence response to stress of the bacterium Bacillus subtilis. Experimental analysis of this behavior via time-lapse fluorescence microscopy, together with mathematical modeling, has allowed us to identify an excitable genetic circuit module that behaves in an excitable way. The study shows that competence for DNA uptake arises in this circuit as a noise-driven excitable state, with the corresponding quiescent state representing vegetative growth. Identification of this excitable module allows us to devise ways to control the excitable dynamics, which could shed light on the evolutionary origin of this phenotypic behavior. As a second example of excitable dynamics in gene regulation, I will discuss the maintenance of pluripotency in mouse embryonic stem cells.

Dynamical insigths on the history-dependence during continuous presentation of rivaling stimuli
Pedro E. GARCÍA (CRM - Univesitat Pompeu Fabra)

Abstract: A recent work tackling the role of adaptation in bi-stable perception has demonstrated that consecutive dominances phases reported during continuous presentation of rivaling stimuli are not statistically independent. The research is based on a novel concept in which a cumulative history for each reported percept during a session is computed by the exponential convolution – with time constant τdecay – of the previous perceptual trace. Then a no monotonic profile for the history-dominance times correlation vs. τdecay is obtained, reaching a significant maximum of 0.2–0.5 at values 0.3< τdecay/Tdom<1 (Tdom - average dominance duration of the complete session). Amazingly, a significant influence of such an integral measure of history on transition times and transition direction is also detected. When the respective cumulative histories of both percepts were balanced, transition durations and the likelihood of return (failed) transitions peaked. We carry out intensive computational simulations of a simple rate model in the adaptation – inhibition strengths variables. Then we checked the capacity of the system to simultaneously fulfill various experimental constraints, such as a Tdom value of the order of seconds and a coefficient of variation (CV) of about 0.5, together with the observed ranges in the history-dependence of dominance times, transitions durations and probability of uncompleted transition phases. It is found that, even when the data related with transitions remains difficult to be reproduced, such a simple model can account for the constraints related with Tdom, CV and the new ones about the correlation data, whiich restrict even more the zone allowed by the previous known data about Tdom and CV. It can be seen that the system should operate in the vicinity of the bifurcation line of the oscillatory and bistable regimes, and mostly inside the bi-stable zone - where noise is indispensable to get alternations between two states representing the possible interpretations. Further validation tests of the model includes the capacity to reproduce the general trends of history-dependencies shown by transition durations, frequency of failed transitions and Tdom time series.

Modeling the dynamics of calcium-triggered cell exocytosis: a Monte Carlo approach
Virginia GONZÁLEZ-VÉLEZ (Univesidad Autónoma Metropolitana)

Abstract: The diffusion of different types of ions in the cytosol of living cells play a key role in their function. In particular, much attention has been paid to the diffusion of Ca2+ in intracellular media and for modeling this process there is a vast number of biological publications which rely on differential methods. The diffusion of calcium in the cytosol is also referred as calcium buffered diffusion. Models to describe buffered diffusion based on differential equations need drastic geometrical simplifications in order to be solved with ease. Depending on the geometrical assumptions considered most differential models lie within two categories: shell models, in which it is assumed that ions enter uniformly over the cell membrane and only the radial diffusion is of relevance (for spherical cells) and microdomain models, which solve the reaction diffusion equations in the vicinity of a channel pore, assuming azimuthal symmetry around the pore; in this last case it is necessary to assume that all channel pores are equidistant. However, it is well known that ions do not enter uniformly given that the channel pores are not regularly spaced. These departures from symmetry may have an important impact on the functionality of the cells. In [1] it was described how the clustering of channels gives rise to calcium profiles which can not be obtained as a superposition of calcium concentrations developed by each individual channel. This non-linearity amplifies the effect of irregular distributions of calcium channels on the formation of submembrane calcium concentration profiles. With the idea in mind to overcome the geometrical simplifications imposed by differential methods, we developed a Monte Carlo simulation for the problem of 3-D buffered calcium diffusion [1,2]. This scheme, which has proved to be successful in the study of the influence of the geometry in the exocytotic response of neuroendocrine cells [3,4], is perfectly suitable for modelling exocytosis in presynaptic terminals, like the Calyx of Held synapse. We discuss resemblances and differences in the modelling of the exocytotic dynamics in both prototype cells.

Exploring the specificity of the relationship between cortical network function and biological simulation parameters with a Particle Swarm Optimization algorithm
David GÓMEZ-CABRERO (IDIBAPS)

Abstract: Mechanistic aspects of brain function can be studied with the use of biologically detailed computational models. Often, a critical question that arises from these computational models is how critically the conclusions depend on a particular choice of the simulation parameters. It is difficult to establish that the presented network model is unique in producing the relevant phenomenology. This issue has been approached before for the case of single neurons or small networks of neurons. Here, we design a computational strategy to explore this for the case of large-scale biological neural network simulations. We focus on a specific neural function: visuo-spatial working memory, and we construct a biological neural network that mimics the cortical network. We search for sets of parameters such that the network sustains persistent activity. We design several evaluation functions that quantify this ability and weigh them in a proper way. To guide the search we rely on the Particle Swarm Optimization. The first objective is to find if there exists a unique solution or a set of significant different solutions. In the second case, we explore and typify the different areas of solutions. Analysis identify different types of solutions separated in the parameter space; this allow us to identify the main parameters associated with each behaviour, and give further insights of the model.

Stochastic neuronal dynamics
Vincent HAKIM (École Normale Supérieure)

Abstract: Neurons operate in vivo under strongly fluctuating inputs. I will introduce important quantities that help to analyze the ensuing stochastic dynamics. I will then show how they can be used to describe correlations between spike trains as well as oscillations in neuronal networks. Recent data on cerebellar cell networks will serve as an illustration.

The use of stochastic differential equations in recovering synchronization dynamics from macroscopic recordings of neuronal activity
Rikkert HINDRIKS (Vreij Universiteit Amsterdam)

Abstract: To study the mechanisms underlying large-scale synchronization processes in the human brain we fit coupled stochastic differential equations directly to human magnetoencephalographic (MEG) recordings. The used model allows for a characterization between neuronal interactions in phase and amplitude as well as between interactions arising from deterministic and stochastic forces. We discuss the general applicability of coupled stochastic differential equations to model EEG/MEG recordings and address the statistical issues that arise. While the use of stochastic differential equations in computational neuroscience is becoming widespread [1] their use in the analysis of macroscopic recordings of neural activity is still very limited. Most currently used EEG/MEG data-analysis methods focus on the detection and quantification of synchronization and do not provide insight into the dynamical mechanisms underlying the measurements. For this purpose, neuronal interaction is modeled explicitly and is directly fitted to the data. While deterministic synchronization models are increasingly used [2] the use of stochastic dynamical models is still very limited [3]. However, since noise is inherent to biophysical systems and given the meanfield nature of EEG/MEG recordings, models with a stochastic component are expected to provide a more realistic description of the electrophysiological processes underlying EEG/MEG recordings. We fit a system of coupled stochastic differential equations to human MEG recordings of healthy subjects and Parkinsonian patients during a finger tapping task, localized to left and right primary motor cortices. We discuss the statistical issues that arise, like, an appropriate method to estimate drift and diffusion coefficients, how to check the goodness of fit, and how to assess the Markov property. The results provide insight into the dynamical mechanisms underlying interaction between primary motor cortices and their degradation as a consequence of Parkinson’s disease.

A Fokker-Planck equation for interacting neurons
Simona MANCINI (Univesité d'Orleans)

Abstract: We are interested in the study of the statistical properties of a large set of interacting neurons. Their behaviour is described in terms of the evolution of respective frequencies, whose dynamics is modelled by coupled stochastic ordinary differential equations, (ods), as done by Deco et al. From this system, we classically obtain a Fokker-Planck equation, describing the evolution in time of the distribution function with respect to the two family frequencies. We shall prove the existence, uniqueness and positivity of the solution at equilbrium for the Fokker- Planck model. Moreover, numerical simulations show that, at the equilibrium, the solution has double gaussian profile, which is the same obtained by Deco et al. by means of moment analysis on the ods. Finally, we can compute the "escaping" time for differents values of the diffusion coefficient, and show that it has an exponential behaviour with respect to the diffusion coefficient.

Axon growth in neural development: sensing, transduction and movement
Giovanni NALDI (Università degli Studi di Milano)

Abstract: In the embryo, undifferentiated sets of cells form organized patterns following pathways marked by chemical cues. At this small scale, cues are represented by single molecules, displaced from their release location by diffusion. Diffusion is the movement of matter from areas with higher concentrations (near the source) to areas of lower concentrations. This works very much like the spreading by the grapevine of a metropolitan legend: news travel at a given speed and are subject to a progressive degradation. Cells crawl along the positive gradient, towards the direction of increasing chemical signal, from the periphery to the source. This establishes the controlled flow of material needed to build structured tissues. We may ask how far from its birthplace can we hear the metropolitan legend. Analogously, how far from its source can a chemical cue be found? The mathematics of diffusion shows that there exists a characteristic maximum reachable distance, called diffusion length, that depends on the volume (or on the weight) of the diffused molecule and on its activity time. Another aspect that we should consider is the fact that in the embryo, very much like in a noisy square, different cues are present at the same time. Following the chemical gradients can thus be as difficult as trying to localize (without a cell phone!) a friend who is calling us, lost in the crowd. How should we look for our friend and reach him? Cells work out the right direction sensing the chemical cues released in the environment, filtering out noise. To understand this mechanism, it is essential to dig into the process of gradient sensing. Cells try to detect very small differences in molecule concentration across their tiny diameter. With this respect, they behave like an instrument that counts molecules in its surroundings and is allowed only a limited number of probings. The study of the measurement errors of such an instrument can explain the shape of the trajectories. Moreover, we know that repeating the measure can reduce uncertainty, but it requires more time. A mathematical model of the measuring process and of the subsequent cell motion sheds light on the balance between the unevenness of trajectories and the time span of the motion in different conditions. This analysis can explain why neurons grow more slowly when the surrounding environment is more complex, for example when they have to perform sharp turns like when they approach the developing spinal cord. The model also suggests that some sort of amplification of the signal must occur inside the cell. This effect stems from a cascade of intracellular biochemical reactions that are only partially known to biologists. Mathematics can predict the magnitude of the amplification needed to separate a weak, but coherent signal, from the background noise and explain how even a couple of molecules in more in a certain direction can make the difference for life.

Analyzing neuronal networks using discrete-time dynamical systems
David Hillel TERMAN (Ohio State University)

Abstract: We describe mathematical techniques for analyzing detailed biophysical models for excitatory-inhibitory neuronal networks. While these networks arise in numerous applications, the focus of this talk will be to better understand mechanisms that underlie temporally dynamic responses in the olfactory system. Our strategy is to first reduce the model to a discrete-time dynamical system. Using the discrete model, we can systematically study how the emergent firing patterns depend on parameters including the underlying network architecture.

Modeling neural networks with activity dependent synapses
Joaquín J. TORRES (Universidad de Granada)

Abstract: In this talk, I review some of our recent work over the effect of activity-dependent synaptic processes, such as short-time depression and facilitation, on the emergent behavior of recurrent neural networks. Depending on the synapse dynamics, different types of behavior can emerge, including a standard recall phase, a novel oscillatory regime, and a non-recall phase. In the oscillatory phase, the activity of the network continously jumps between different attractors associated to previously stored patterns. Our study have shown that the interplay between synaptic depression and stochasticity is important for destabilizing the attractors. This property is enhanced by synaptic facilitation which, therefore, improves the adaptation of the system to external stimuli. A detailed analysis of the network also reflects an efficient (more rapid and with less error) access to the memories when facilitation is increased. We also investigated the influence of facilitation and depression on the maximum storage capacity of the network. Our study shows that synaptic depression drastically reduces the capacity of the system to store an retrieve memories. Facilitation, however, enhances the memory capacity in different situations. In particular, we found optimal values of the relevant synaptic parameters for which the storage capacity can be maximal and similar to the one obtained with “static” synapses. We conclude that depressing synapses with a certain level of facilitation allow to recover the good retrieval properties of neural networks with static synapses while maintaining the nonlinear properties of dynamic synapses, convenient for dynamical information processing and coding.

Fluid limit theorems for stochastic hybrid systems with application to neuron models
Gilles WAINRIB (Université Paris VII)

Abstract: Neurons are subject to various source of noise, both extrinsic and intrinsic. The main source of intrinsic noise is the stochastic behaviour of ion channels. Within the framework of stochastic hybrid processes, we establish mathematical results describing the limit of large number of ion channels.