Shared inputs, entrainment, and desynchrony in elliptic bursters: from slow passage to discontinuous circle maps

What input signals will lead to synchrony vs. desynchrony in a group of biological oscillators? This question connects with both classical dynamical systems analyses of entrainment and phase locking and with emerging studies of stimulation patterns for controlling neural network activity. Here, we focus on the response of a population of uncoupled, elliptically bursting neurons to a common pulsatile input. We extend a phase reduction from the literature to capture inputs of varied strength, leading to a circle map with discontinuities of various orders. In a combined analytical and numerical approach, we apply our results to both a normal form model for elliptic bursting and to a biophysically-based neuron model from the basal ganglia. We find that, depending on the period and amplitude of inputs, the response can either appear chaotic (with provably positive Lyaponov exponent for the associated circle maps), or periodic with a broad range of phase-locked periods. Throughout, we discuss the critical underlying mechanisms, including slow-passage effects through Hopf bifurcation, the role and origin of discontinuities, and the impact of noiseComment: 17 figures, 40 page

How single neuron properties shape chaotic dynamics and signal transmission in random neural networks

While most models of randomly connected networks assume nodes with simple dynamics, nodes in realistic highly connected networks, such as neurons in the brain, exhibit intrinsic dynamics over multiple timescales. We analyze how the dynamical properties of nodes (such as single neurons) and recurrent connections interact to shape the effective dynamics in large randomly connected networks. A novel dynamical mean-field theory for strongly connected networks of multi-dimensional rate units shows that the power spectrum of the network activity in the chaotic phase emerges from a nonlinear sharpening of the frequency response function of single units. For the case of two-dimensional rate units with strong adaptation, we find that the network exhibits a state of "resonant chaos", characterized by robust, narrow-band stochastic oscillations. The coherence of stochastic oscillations is maximal at the onset of chaos and their correlation time scales with the adaptation timescale of single units. Surprisingly, the resonance frequency can be predicted from the properties of isolated units, even in the presence of heterogeneity in the adaptation parameters. In the presence of these internally-generated chaotic fluctuations, the transmission of weak, low-frequency signals is strongly enhanced by adaptation, whereas signal transmission is not influenced by adaptation in the non-chaotic regime. Our theoretical framework can be applied to other mechanisms at the level of single nodes, such as synaptic filtering, refractoriness or spike synchronization. These results advance our understanding of the interaction between the dynamics of single units and recurrent connectivity, which is a fundamental step toward the description of biologically realistic network models in the brain, or, more generally, networks of other physical or man-made complex dynamical units

Dynamics meets Morphology: towards Dymorph Computation

In this dissertation, approaches are presented for both technically using and investigating biological principles with oscillators in the context of electrical engineering, in particular neuromorphic engineering. Thereby, dynamics as well as morphology as important neuronal principles were explicitly selected, which shape the information processing in the human brain and distinguish it from other technical systems. The aspects and principles selected here are adaptation during the encoding of stimuli, the comparatively low signal transmission speed, the continuous formation and elimination of connections, and highly complex, partly chaotic, dynamics. The selection of these phenomena and properties has led to the development of a sensory unit that is capable of encoding mechanical stress into a series of voltage pulses by the use of a MOSFET augmented by AlScN. The circuit is based on a leaky integrate and fire neuron model and features an adaptation of the pulse frequency. Furthermore, the slow signal transmission speed of biological systems was the motivation for the investigation of a temporal delay in the feedback of the output pulses of a relaxation oscillator. In this system stable pulse patterns which form due to so-called jittering bifurcations could be observed. In particular, switching between different stable pulse patterns was possible to induce. In the further course of the work, the first steps towards time-varying coupling of dynamic systems are investigated. It was shown that in a system consisting of dimethyl sulfoxid and zinc acetate, oscillators can be used to force the formation of filaments. The resulting filaments then lead to a change in the dynamics of the oscillators. Finally, it is shown that in a system with chaotic dynamics, the extension of it with a memristive device can lead to a transient stabilisation of the dynamics, a behaviour that can be identified as a repeated pass of Hopf bifurcations

Mean-Field Coupled Systems and Self-Consistent Transfer Operators: A Review

In this review we survey the literature on mean-field coupled maps. We start with the early works from the physics literature, arriving to some recent results from ergodic theory studying the thermodynamic limit of globally coupled maps and the associated self-consistent transfer operators. We also give few pointers to related research fields dealing with mean-field coupled systems in continuous time, and applications

Existence of Mutual Stabilization in Chaotic Neural Models

Recent work has demonstrated that interacting chaotic systems can establish persistent, periodic behavior, called mutual stabilization, when certain information is passed through interaction functions. In particular, this was first shown with two interacting cupolets (Chaotic Unstable Periodic Orbit-lets) of the double scroll oscillator. Cupolets are highly accurate approximations of unstable periodic orbits of a chaotic attractor that can be generated through a control scheme that repeatedly applies perturbations along Poincaré sections. The decision to perturb or not to perturb the trajectory is determined by a bit in a binary control sequence. One interaction function used in the original cupolet research was based on integrate-and-fire dynamics that are often seen in neural and laser systems and was used to demonstrate mutual stabilization between two double scroll oscillators. This result provided the motivation for this thesis where the stabilization of chaos in mathematical models of communicating neurons is investigated. This thesis begins by introducing mathematical models of neurons and discusses the biological realism of the models. Then, we consider the two-dimensional FitzHugh-Nagumo (FHN) neural model and we show how two FHN neurons can exhibit chaotic behavior when communication is mediated by a coupling constant, g, representative of the synaptic strength between the neurons. Through a bifurcation analysis, where the synaptic strength is the bifurcation parameter, we analyze the space of possible long-term behaviors of this model. After identifying regions of periodic and chaotic behavior, we show how a synaptic sigmoidal learning rule transitions the chaotic dynamics of the system to periodic dynamics in the presence of an external signal. After the signal passes through the synapse, synaptic learning alters the synaptic strength and the two neurons remain in a persistent, mutually stabilized periodic state even after the signal is removed. This result provides a proof-of-concept for chaotic stabilization in communicating neurons. Next, we focus on the 3-dimensional Hindmarsh-Rose (HR) neural model that is known to exhibit chaotic behavior and bursting neural firing. Using this model, we create a control scheme using two Poincaré sections in a manner similar to the control scheme for the double scroll system. Using the control scheme we establish that it is possible to generate cupolets in the HR model. We use the HR model to create neural networks where the communication between neurons is mediated by an integrate-and-fire interaction function. With this interaction, we show how a signal can propagate down a unidirectional chain of chaotic neurons. We further show how mutual stabilization can occur if two neurons communicate through this interaction function. Lastly, we expand the investigation to more complicated networks including a feedback network and a chain of neurons that ends in a feedback loop between the two terminal neurons. Mutual stabilization is found to exist in all cases. At each stage, we comment on the potential biological implications and extensions of these results

Loss of synchrony in an inhibitory network of type-I oscillators

Synchronization of excitable cells coupled by reciprocal inhibition is a topic of significant interest due to the important role that inhibitory synaptic interaction plays in the generation and regulation of coherent rhythmic activity in a variety of neural systems. While recent work revealed the synchronizing influence of inhibitory coupling on the dynamics of many networks, it is known that strong coupling can destabilize phase-locked firing. Here we examine the loss of synchrony caused by an increase in inhibitory coupling in networks of type-I Morris-Lecar model oscillators, which is characterized by a period-doubling cascade and leads to mode-locked states with alternation in the firing order of the two cells, as reported recently by Maran and Canavier (2007) for a network of Wang-Buzsáki model neurons. Although alternating- order firing has been previously reported as a near-synchronous state, we show that the stable phase difference between the spikes of the two Morris-Lecar cells can constitute as much as 70% of the unperturbed oscillation period. Further, we examine the generality of this phenomenon for a class of type-I oscillators that are close to their excitation thresholds, and provide an intuitive geometric description of such leap-frog dynamics. In the Morris-Lecar model network, the alternation in the firing order arises under the condition of fast closing of K+ channels at hyperpolarized potentials, which leads to slow dynamics of membrane potential upon synaptic inhibition, allowing the presynaptic cell to advance past the postsynaptic cell in each cycle of the oscillation. Further, we show that non-zero synaptic decay time is crucial for the existence of leap-frog firing in networks of phase oscillators. However, we demonstrate that leap-frog spiking can also be obtained in pulse-coupled inhibitory networks of one-dimensional oscillators with a multi-branched phase domain, for instance in a network of quadratic integrate-and-fire model cells. Also, we show that the entire bifurcation structure of the network can be explained by a simple scaling of the STRC (spike- time response curve) amplitude, using a simplified quadratic STRC as an example, and derive the general conditions on the shape of the STRC function that leads to leap-frog firing. Further, for the case of a homogeneous network, we establish quantitative conditions on the phase resetting properties of each cell necessary for stable alternating-order spiking, complementing the analysis of Goel and Ermentrout (2002) of the order-preserving phase transition map. We show that the extension of STRC to negative values of phase is necessary to predict the response of a model cell to several close non-weak perturbations. This allows us for instance to accurately describe the dynamics of non-weakly coupled network of three model cells. Finally, the phase return map is also extended to the heterogenous network, and is used to analyze both the order-alternating firing and the order-preserving non-zero phase locked state in this case

Predicting single spikes and spike patterns with the Hindmarsh-Rose model

Most simple neuron models are only able to model traditional spiking behavior. As physiologists discover and classify different electrical phenotypes, computational neuroscientists become interested in using simple phenomenological models that can exhibit these different types of spiking patterns. The Hindmarsh-Rose model is a three-dimensional relaxation oscillator which can show both spiking and bursting patterns and has a chaotic regime. We test the predictive powers of the Hindmarsh-Rose model on two different test databases. We show that the Hindmarsh-Rose model can predict the spiking response of rat layer 5 neocortical pyramidal neurons on a stochastic input signal with a precision comparable to the best known spiking models. We also show that the Hindmarsh-Rose model can capture qualitatively the electrical footprints in a database of different types of neocortical interneurons. When the model parameters are fit from sub-threshold measurements only, the model still captures well the electrical phenotype, which suggests that the sub-threshold signals contain information about the firing patterns of the different neuron

Dynamical systems techniques in the analysis of neural systems

As we strive to understand the mechanisms underlying neural computation, mathematical models are increasingly being used as a counterpart to biological experimentation. Alongside building such models, there is a need for mathematical techniques to be developed to examine the often complex behaviour that can arise from even the simplest models. There are now a plethora of mathematical models to describe activity at the single neuron level, ranging from one-dimensional, phenomenological ones, to complex biophysical models with large numbers of state variables. Network models present even more of a challenge, as rich patterns of behaviour can arise due to the coupling alone. We first analyse a planar integrate-and-fire model in a piecewise-linear regime. We advocate using piecewise-linear models as caricatures of nonlinear models, owing to the fact that explicit solutions can be found in the former. Through the use of explicit solutions that are available to us, we categorise the model in terms of its bifurcation structure, noting that the non-smooth dynamics involving the reset mechanism give rise to mathematically interesting behaviour. We highlight the pitfalls in using techniques for smooth dynamical systems in the study of non-smooth models, and show how these can be overcome using non-smooth analysis. Following this, we shift our focus onto the use of phase reduction techniques in the analysis of neural oscillators. We begin by presenting concrete examples showcasing where these techniques fail to capture dynamics of the full system for both deterministic and stochastic forcing. To overcome these failures, we derive new coordinate systems which include some notion of distance from the underlying limit cycle. With these coordinates, we are able to capture the effect of phase space structures away from the limit cycle, and we go on to show how they can be used to explain complex behaviour in typical oscillatory neuron models