Connectivity Kernels Research Articles

Pattern formation in a 2-population homogenized neuronal network model

We study pattern formation in a 2-population homogenized neural field model of the Hopfield type in one spatial dimension with periodic microstructure. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. It is shown that the nonlocal synaptic interactions promote a finite band width instability. The stability method relies on a sequence of wave-number dependent invariants of 2times 2-stability matrices representing the sequence of Fourier-transformed linearized evolution equations for the perturbation imposed on the homogeneous background. The generic picture of the instability structure consists of a finite set of well-separated gain bands. In the shallow firing rate regime the nonlinear development of the instability is determined by means of the translational invariant model with connectivity kernels replaced with the corresponding period averaged connectivity functions. In the steep firing rate regime the pattern formation process depends sensitively on the spatial localization of the connectivity kernels: For strongly localized kernels this process is determined by the translational invariant model with period averaged connectivity kernels, whereas in the complementary regime of weak and moderate localization requires the homogenized model as a starting point for the analysis. We follow the development of the instability numerically into the nonlinear regime for both steep and shallow firing rate functions when the connectivity kernels are modeled by means of an exponentially decaying function. We also study the pattern forming process numerically as a function of the heterogeneity parameters in four different regimes ranging from the weakly modulated case to the strongly heterogeneous case. For the weakly modulated regime, we observe that stable spatial oscillations are formed in the steep firing rate regime, whereas we get spatiotemporal oscillations in the shallow regime of the firing rate functions.

Inference of a universal social scale and segregation measures using social connectivity kernels.

How people connect with one another is a fundamental question in the social sciences, and the resulting social networks can have a profound impact on our daily lives. Blau offered a powerful explanation: people connect with one another based on their positions in a social space. Yet a principled measure of social distance, allowing comparison within and between societies, remains elusive. We use the connectivity kernel of conditionally independent edge models to develop a family of segregation statistics with desirable properties: they offer an intuitive and universal characteristic scale on social space (facilitating comparison across datasets and societies), are applicable to multivariate and mixed node attributes, and capture segregation at the level of individuals, pairs of individuals and society as a whole. We show that the segregation statistics can induce a metric on Blau space (a space spanned by the attributes of the members of society) and provide maps of two societies. Under a Bayesian paradigm, we infer the parameters of the connectivity kernel from 11 ego-network datasets collected in four surveys in the UK and USA. The importance of different dimensions of Blau space is similar across time and location, suggesting a macroscopically stable social fabric. Physical separation and age differences have the most significant impact on segregation within friendship networks with implications for intergenerational mixing and isolation in later stages of life.

From receptive profiles to a metric model of V1.

In this work we show how to construct connectivity kernels induced by the receptive profiles of simple cells of the primary visual cortex (V1). These kernels are directly defined by the shape of such profiles: this provides a metric model for the functional architecture of V1, whose global geometry is determined by the reciprocal interactions between local elements. Our construction adapts to any bank of filters chosen to represent a set of receptive profiles, since it does not require any structure on the parameterization of the family. The connectivity kernel that we define carries a geometrical structure consistent with the well-known properties of long-range horizontal connections in V1, and it is compatible with the perceptual rules synthesized by the concept of association field. These characteristics are still present when the kernel is constructed from a bank of filters arising from an unsupervised learning algorithm.

Stability and Spatiotemporal Bifurcations in Spatially Distributed Neural Networks with Nonlocal Delay

Abstract The stability of equilibria and bifurcations of neural networks in a real line with nonlocal delay are presented. A sufficient condition of stable equilibria is declared by the linear part. Eigenvalue analysis implies the existence of bifurcations, and by exploiting typical excitatory and inhibitory connectivity kernels in a neural network, the possible bifurcations are discussed according to various cases. It is an advantageous tool using a multiple-scale method to study the stability of bifurcated travelling waves or spots. As an illustration of our theory, the dynamics of a seashell continuous-time circular mask model are investigated. It is shown that both the shape and range of active function and synaptic weights can affect the dynamics of the model. Finally, the bifurcation set and the variety of bifurcated patterns of the seashell model are numerically revealed.

Local and Global Gestalt Laws: A Neurally Based Spectral Approach.

This letter presents a mathematical model of figure-ground articulation that takes into account both local and global gestalt laws and is compatible with the functional architecture of the primary visual cortex (V1). The local gestalt law of good continuation is described by means of suitable connectivity kernels that are derived from Lie group theory and quantitatively compared with long-range connectivity in V1. Global gestalt constraints are then introduced in terms of spectral analysis of a connectivity matrix derived from these kernels. This analysis performs grouping of local features and individuates perceptual units with the highest salience. Numerical simulations are performed, and results are obtained by applying the technique to a number of stimuli.

A one-population Amari model with periodic microstructure

We review the derivation of the homogenized one-population Amari equation by means of the two-scale convergence technique of Nguetseng in the case of periodic microvariation in the connectivity function. A key point in this derivation is Visintin's theorem for two-scale convergence of convolution integrals. We construct single bump solutions of the resulting homogenized equation using a pinning function technique for the case where the solutions are independent of the local variable and the firing rate function is modelled as a unit step function. The parameter measuring the degree of heterogeneity plays the role of a control parameter. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. A framework for analysing the stability of these structures is formulated. This framework is based on spectral theory for Hilbert–Schmidt integral operators and it deforms to the standard Evans function approach for the translational invariant case in the limit of no heterogeneity. The upper and lower bounds of the growth/decay rates of the perturbations imposed on the bump states can be expressed in terms of the operator norm of the actual Hilbert–Schmidt operator. Intervals for which the pinning function is increasing correspond to unstable bumps, while complementary intervals where the pinning function decreases correspond to stable bumps, just as in the translational invariant case. Examples showing the properties of the bumps are discussed in detail when the connectivity kernels are given in terms of an exponential decaying function, a wizard hat function and a damped oscillating function.

Parametric Anatomical Modeling: A method for modeling the anatomical layout of neurons and their projections

Event Abstract Back to Event Parametric Anatomical Modeling: A method for modeling the anatomical layout of neurons and their projections Martin Pyka1* and Sen Cheng1 1 Ruhr-University Bochum, Department of Psychiatry, Germany Biological neural networks are likely to be described by a low-dimensional parameter space. Those parameters include, for example, the 3d-shape of neuron layers, the neurons' spatial projection patterns, spiking dynamics and plasticity rules. Many studies generate artificial neural networks that match some of the properties of biological networks to study their computational properties. While spiking dynamics and plasticity rules can comparatively easily be expressed in a programming language and have been subject of extensive research, defining and investigating anatomically more realistic connection and latency properties for large-scale artificial neural networks remains a challenging task. In particular, a method is missing that allows to translate anatomical data from e.g. histological images and tracer studies into an encoding that can generate more realistic network architectures. We present a new method, called Parametric Anatomical Modeling (PAM), to fill this gap. The basic idea behind PAM is that neural networks are computed based on neuronal, synaptic and intermediate layers, which are generated from anatomical data. Using a set of mapping techniques, complex connection patterns between those layers can be easily defined (fig. 1). For example, any location on one layer can be mapped on another layer based on euclidean distance, a normal vector or topological similarity. Connections between neurons are computed by mapping pre- and post-synaptic locations on a synaptic layer and applying connectivity kernels on the surface of the synaptic layer. A central feature of PAM is that distances between neurons (which may affect transmission delays) can be computed by combining spatial distances on the surface of layers and spatial distances between layers. Thereby, complex connection and distance patterns between layers can be expressed and be used as template for generating small-scale and large-scale network architectures. PAM is implemented as a Python tool and integrated in the 3d-modeling software Blender. We provide a set of add-ons and Python modules that amend the functionality of Blender to generate and relate anatomical layers to each other and to create neural networks for the network simulator NEST. These tools along with example files and video tutorials will be freely available. On a 3d-model of the hippocampus and entorhinal cortex, we demonstrate the benefits of PAM and show how PAM can help to uncover the relationship between the form and function of the hippocampus. Models created by PAM can also serve as an educational tool to visualize the 3-d connectivity of brain regions. The low-dimensional, but yet biologically plausible, parameter space renders PAM particularly suitable for analysing allometric and evolutionary factors in networks and for modeling the complexity of real networks with comparatively little effort. Figure 1 Keywords: Hippocampus, 3D model, consolidation, evolution, Blender, anatomical modeling Conference: Neuroinformatics 2014, Leiden, Netherlands, 25 Aug - 27 Aug, 2014. Presentation Type: Demo, to be considered for oral presentation Topic: Large-scale modeling Citation: Pyka M and Cheng S (2014). Parametric Anatomical Modeling: A method for modeling the anatomical layout of neurons and their projections. Front. Neuroinform. Conference Abstract: Neuroinformatics 2014. doi: 10.3389/conf.fninf.2014.18.00020 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 04 Apr 2014; Published Online: 04 Jun 2014. * Correspondence: Dr. Martin Pyka, Ruhr-University Bochum, Department of Psychiatry, Bochum, Germany, martin.pyka@gmx.de Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Martin Pyka Sen Cheng Google Martin Pyka Sen Cheng Google Scholar Martin Pyka Sen Cheng PubMed Martin Pyka Sen Cheng Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Absolute Stability and Complete Synchronization in a Class of Neural Fields Models

Neural fields are an interesting option for modeling macroscopic parts of the cortex involving several populations of neurons, like cortical areas. Two classes of neural field equations are considered: voltage- and activity-based. The spatio-temporal behavior of these fields is described by nonlinear integro-differential equations. The integral term, computed over a compact subset of $\mathbb{R}^q$, $q=1,2,3$, involves space and time varying, possibly nonsymmetric, intracortical connectivity kernels. Contributions from white matter afferents are represented as external input. Sigmoidal nonlinearities arise from the relation between average membrane potentials and instantaneous firing rates. Using methods of functional analysis, we characterize the existence and uniqueness of a solution of these equations for general, homogeneous (i.e., independent of the spatial variable), and spatially locally homogeneous inputs. In all cases we give sufficient conditions on the connectivity functions for the solutions to be absolutely stable, that is to say, asymptotically independent of the initial state of the field. These conditions bear on some compact operators defined from the connectivity kernels, the maximal slope of the sigmoids, and the time constants used in describing the temporal shape of the postsynaptic potentials. Numerical experiments are presented to illustrate the theory. To our knowledge this is the first time that such a complete analysis of the problem of the existence and uniqueness of a solution of these equations has been obtained. Another important contribution is the analysis of the absolute stability of these solutions—more difficult but more general than the linear stability analysis which it implies. The reason we have been able to complete this work is our use of the functional analysis framework and the theory of compact operators in a Hilbert space which has allowed us to provide simple mathematical answers to some of the questions raised by modelers in neuroscience.

Neural Fields with Distributed Transmission Speeds and Long‐Range Feedback Delays

We introduce distributed axonal transmission speeds and a long‐range constant feedback loop into the standard neural field model. We analyze the stability of spatially homogeneous equilibrium solutions for general connectivity kernels. By studying reduced models based on the assumption of small delays, we determine the effects of the delays on the stability and bifurcations. We show in a reduced model that delayed excitatory feedback generally facilitates stationary bifurcations and Turing patterns, while suppressing the bifurcation of periodic solutions and traveling waves. The reverse conclusion holds for inhibitory feedback. In case of oscillatory bifurcations, the variance of the distributed propagation and feedback delays affects the frequency of periodic solutions and the phase speed of traveling waves. Moreover, we give a nonlinear analysis of traveling fronts and find that distributed transmission speeds can maximize the front speed.

Spontaneous and evoked activity in extended neural populations with gamma-distributed spatial interactions and transmission delay

Abstract The spatiotemporal dynamics of neural activity are studied using an integro-differential model of spatially extended neuronal ensembles. The model includes both synaptic and axonal propagation delay while spatial synaptic connectivities are represented by gamma distributions. This family of connectivity kernels has been observed experimentally and covers the cases of divergent, finite, and negligible self-connections. We give conditions for stationary and non-stationary instabilities for gamma-distributed kernels, which can be formulated in terms of the mean spatial interaction ranges and the mean spatial interaction times. We present novel mechanisms for Turing patterns and traveling waves, which result from the special shape of the gamma-distributed interactions. We give a numerical study of the propagation of evoked spatiotemporal response activity caused by short local stimuli, and reveal maximum response activity after the mean interaction time. This maximum occurs at a distance from stimulus offset location, which is equal to the mean interaction range.

Effects of nonlocal feedback on traveling fronts in neural fields subject to transmission delay

The work introduces a model for reciprocal connections in neural fields by a nonlocal feedback mechanism, while the neural field exhibits nonlocal interactions and intra-areal transmission delays. We study the speed of traveling fronts with respect to the transmission delay, the spatial feedback range, and the feedback delay for general axonal and feedback connectivity kernels. In addition, we find a novel shape of traveling fronts due to the applied feedback and criteria for its occurrence are derived.

Stability and Bifurcations in Neural Fields with Finite Propagation Speed and General Connectivity

A stability analysis is presented for neural field equations in the presence of finite propagation speed along axons and for a general class of connectivity kernels and synaptic properties. Sufficient conditions are given for the stability of equilibrium solutions. It is shown that the propagation delays play a significant role in nonstationary bifurcations of equilibria, whereas the stationary bifurcations depend only on the connectivity kernel. In the case of nonstationary bifurcations, bounds are determined on the frequencies of the resulting oscillatory solutions. A perturbative scheme is used to calculate the types of bifurcations leading to spatial patterns, oscillations, and traveling waves. For high propagation speeds a simple method is derived that allows the determination of the bifurcation type by visual inspection of the Fourier transforms of the kernel and its first moment. Results are numerically illustrated on a class of neurologically plausible systems with combinations of Gaussian excitatory and inhibitory connections.

Nonlinear analysis of spatio-temporal receptive fields: IV. Generic tuning properties for rectifying rate-functions

Neural field models have been used to explain tuning properties of V1 neurons. I here show that these models generically lead to contrast-independent tuning if they only have semilinear, zero-threshold rate functions. Other model details are unimportant, as the number and dimension of layers, imprinted feature maps, the precise form of input profiles or connectivity kernels, etc. Furthermore, also spatio-temporal solutions scale linearly with input strength but are otherwise form-invariant.