Stability and Hopf Bifurcation Analysis of Hopfield Neural Networks with a General Distribution of Delays

We investigate the linear stability and perform the bifurcation analysis for Hopfield neural networks with a general distribution of delays, where the neurons are identical. We start by analyzing the scalar model and show what kind of information can be gained with only minimal information about the exact distribution of delays. We determine a mean delay and distribution independent stability region. We then illustrate a way of improving on this conservative result by approximating the true region of stability when the actual distribution is not known, but some moments or cumulants of the distribution are. We compare the approximate stability regions with the stability regions in the case of the uniform and gamma distributions. We show that, in general, the approximations improve as more moments or cumulants are used, and that the approximations using cumulants give better results than the ones using moments. Further, we extend these results to a network of n identical neurons, where we examine the stability of a symmetrical equilibrium point via the analysis of the characteristic equation both when the connection matrix is symmetric and when it is not. Finally, for the scalar model, we show under what conditions a Hopf bifurcation occurs and we use the centre manifold technique to determine the criticality of the bifurcation. When the kernel represents the gamma distribution with p=1 and p=2, we transform the delay differential equation into a system of ordinary differential equations and we compare the centre manifold computation to the one we obtain in the ordinary differential case

Dynamics of neural systems with discrete and distributed time delays

In real-world systems, interactions between elements do not happen instantaneously, due to the time required for a signal to propagate, reaction times of individual elements, and so forth. Moreover, time delays are normally nonconstant and may vary with time. This means that it is vital to introduce time delays in any realistic model of neural networks. In order to analyze the fundamental properties of neural networks with time-delayed connections, we consider a system of two coupled two-dimensional nonlinear delay differential equations. This model represents a neural network, where one subsystem receives a delayed input from another subsystem. An exciting feature of the model under consideration is the combination of both discrete and distributed delays, where distributed time delays represent the neural feedback between the two subsystems, and the discrete delays describe the neural interaction within each of the two subsystems. Stability properties are investigated for different commonly used distribution kernels, and the results are compared to the corresponding results on stability for networks with no distributed delays. It is shown how approximations of the boundary of the stability region of a trivial equilibrium can be obtained analytically for the cases of delta, uniform, and weak gamma delay distributions. Numerical techniques are used to investigate stability properties of the fully nonlinear system, and they fully confirm all analytical findings

Learning Image-Conditioned Dynamics Models for Control of Under-actuated Legged Millirobots

Millirobots are a promising robotic platform for many applications due to their small size and low manufacturing costs. Legged millirobots, in particular, can provide increased mobility in complex environments and improved scaling of obstacles. However, controlling these small, highly dynamic, and underactuated legged systems is difficult. Hand-engineered controllers can sometimes control these legged millirobots, but they have difficulties with dynamic maneuvers and complex terrains. We present an approach for controlling a real-world legged millirobot that is based on learned neural network models. Using less than 17 minutes of data, our method can learn a predictive model of the robot's dynamics that can enable effective gaits to be synthesized on the fly for following user-specified waypoints on a given terrain. Furthermore, by leveraging expressive, high-capacity neural network models, our approach allows for these predictions to be directly conditioned on camera images, endowing the robot with the ability to predict how different terrains might affect its dynamics. This enables sample-efficient and effective learning for locomotion of a dynamic legged millirobot on various terrains, including gravel, turf, carpet, and styrofoam. Experiment videos can be found at https://sites.google.com/view/imageconddy

Stationary bumps in a piecewise smooth neural field model with synaptic depression

We analyze the existence and stability of stationary pulses or bumps in a one–dimensional piecewise smooth neural field model with synaptic depression. The continuum dynamics is described in terms of a nonlocal integrodifferential equation, in which the integral kernel represents the spatial distribution of synaptic weights between populations of neurons whose mean firing rate is taken to be a Heaviside function of local activity. Synaptic depression dynamically reduces the strength of synaptic weights in response to increases in activity. We show that in the case of a Mexican hat weight distribution, there exists a stable bump for sufficiently weak synaptic depression. However, as synaptic depression becomes stronger, the bump became unstable with respect to perturbations that shift the boundary of the bump, leading to the formation of a traveling pulse. The local stability of a bump is determined by the spectrum of a piecewise linear operator that keeps track of the sign of perturbations of the bump boundary. This results in a number of differences from previous studies of neural field models with Heaviside firing rate functions, where any discontinuities appear inside convolutions so that the resulting dynamical system is smooth. We also extend our results to the case of radially symmetric bumps in two–dimensional neural field models

A physical model suggests that hip-localized balance sense in birds improves state estimation in perching: implications for bipedal robots

In addition to a vestibular system, birds uniquely have a balance-sensing organ within the pelvis, called the lumbosacral organ (LSO). The LSO is well developed in terrestrial birds, possibly to facilitate balance control in perching and terrestrial locomotion. No previous studies have quantified the functional benefits of the LSO for balance. We suggest two main benefits of hip-localized balance sense: reduced sensorimotor delay and improved estimation of foot-ground acceleration. We used system identification to test the hypothesis that hip-localized balance sense improves estimates of foot acceleration compared to a head-localized sense, due to closer proximity to the feet. We built a physical model of a standing guinea fowl perched on a platform, and used 3D accelerometers at the hip and head to replicate balance sense by the LSO and vestibular systems. The horizontal platform was attached to the end effector of a 6 DOF robotic arm, allowing us to apply perturbations to the platform analogous to motions of a compliant branch. We also compared state estimation between models with low and high neck stiffness. Cross-correlations revealed that foot-to-hip sensing delays were shorter than foot-to-head, as expected. We used multi-variable output error state-space (MOESP) system identification to estimate foot-ground acceleration as a function of hip- and head-localized sensing, individually and combined. Hip-localized sensors alone provided the best state estimates, which were not improved when fused with head-localized sensors. However, estimates from head-localized sensors improved with higher neck stiffness. Our findings support the hypothesis that hip-localized balance sense improves the speed and accuracy of foot state estimation compared to head-localized sense. The findings also suggest a role of neck muscles for active sensing for balance control: increased neck stiffness through muscle co-contraction can improve the utility of vestibular signals. Our engineering approach provides, to our knowledge, the first quantitative evidence for functional benefits of the LSO balance sense in birds. The findings support notions of control modularity in birds, with preferential vestibular sense for head stability and gaze, and LSO for body balance control,respectively. The findings also suggest advantages for distributed and active sensing for agile locomotion in compliant bipedal robots

Integration of continuous-time dynamics in a spiking neural network simulator

Contemporary modeling approaches to the dynamics of neural networks consider two main classes of models: biologically grounded spiking neurons and functionally inspired rate-based units. The unified simulation framework presented here supports the combination of the two for multi-scale modeling approaches, the quantitative validation of mean-field approaches by spiking network simulations, and an increase in reliability by usage of the same simulation code and the same network model specifications for both model classes. While most efficient spiking simulations rely on the communication of discrete events, rate models require time-continuous interactions between neurons. Exploiting the conceptual similarity to the inclusion of gap junctions in spiking network simulations, we arrive at a reference implementation of instantaneous and delayed interactions between rate-based models in a spiking network simulator. The separation of rate dynamics from the general connection and communication infrastructure ensures flexibility of the framework. We further demonstrate the broad applicability of the framework by considering various examples from the literature ranging from random networks to neural field models. The study provides the prerequisite for interactions between rate-based and spiking models in a joint simulation