Nonlinear Dynamics in Computational Neuroscience

Neural mass models have been actively used since the 1970s to model the coarse grained activity of large populations of neurons and synapses. They have proven especially useful in understanding brain rhythms. However, although motivated by neurobiological considerations they are phenomenological in nature, and cannot hope to recreate some of the rich repertoire of responses seen in real neuronal tissue. In this chapter we consider the \(\theta \)-neuron model that has recently been shown to admit to an exact mean-field description for instantaneous pulsatile interactions. We show that the inclusion of a more realistic synapse model leads to a mean-field model that has many of the features of a neural mass model coupled to a further dynamical equation that describes the evolution of network synchrony. A bifurcation analysis is used to uncover the primary mechanism for generating oscillations at the single and two population level. Numerical simulations also show that the phenomena of event related synchronisation and desynchronisation are easily realised. Importantly unlike its phenomenological counterpart this next generation neural mass model is an exact macroscopic description of an underlying microscopic spiking neurodynamics, and is a natural candidate for use in future large scale human brain simulations.

The intricate web of connections among the neurons composing the cerebral cortex is the seed of the complexity that our brain is capable to express. Such complexity is organized as it results from a hierarchical and modular organization of the network in which the roles of different cortical areas are distinct. Here, we speculate that such differentiation can be obtained by relying on the granular nature of the cortical surface tiled with ‘canonic’ modules which in turn can be flexibly tuned to compose diverse mesoscopic networks. The remarkable versatility of these cortical modules is governed by few key parameters like the excitability level and the sensitivity to the accumulated activity-dependent fatigue. These modules are naturally endowed with a rich repertoire of activity regimes which range from quasi-stable dynamics, possibly exploited to store information or provide persistent input to other modules, to collective oscillations reminiscent of the Up/Down activity cycle observed during sleep and deep anesthesia. Finally, we conclude showing that such slow oscillations, spontaneously expressed by the isolated cortex, can provide an ideal experimental framework to infer the dynamical properties of these cortical modules which in turn can inform also on cortical function in other brain states, such as during wakefulness.

The capacity of the brain to change its basic functions and structures is called neural plasticity. Neural plasticity is one of the most challenging themes in neuroscience and its comprehension may lead to fundamental understanding on brain dynamics. Here, we have characterized the intra and inter regional neuroplastic changes in animal models (mice) of different diseases (i.e. stroke and epilepsy) and sensorimotor stimulation induced by environmental enrichment conditions through the quantitative linear and nonlinear analysis of electrophysiological signals (Local Field Potentials, LFPs). Various properties characterizing LFPs such as power spectra, scaling behavior and interdependence have been quantified. These characterizations were able to discriminate between different experimental conditions, thus providing a good set of quantities that could be useful as biomarkers in medical diagnostics. In particular, we reported some cases in which nonlinear time series analysis reveals effects that are not detected by linear methods. For the epileptic mice, the spectral analysis has shown that epileptic activity determines a power redistribution among the different neurophysiological bands. Symbolic Transfer Entropy measure indicates a greater driving influence of the focal epileptic side on activity in the contra-lateral hemisphere, while Granger causality measures fails at detecting it. Inter-hemispheric functional coupling within delta band (0.5–4 Hz) was reduced in homotopic Pre-Motor Areas of ischemic animals, as shown by a statistically significant decrease in the mutual information measures (not captured by cross-correlation index). Finally, we estimated the scaling properties of LFPs recorded from freely-moving mice reared in environmental enrichment (EE) and standard condition (SC) by using an integrated approach combining three different techniques: the Higuchi method, DFA and spectral analysis. Our results indicated the existence of a particular power spectrum scaling law \(1/f^{\beta }\) with \(\beta < 0\) for low frequencies (\(f< 4 \,\)Hz) for both SC and EE rearing conditions. Notably, the difference between scaling exponents in EE and SC for individual cortical regions (M2) and (V1) was not statistically significant. Altogether, these findings shed light on the mechanism involved in neocortical plasticity suggesting both robust plasticity of transcallosal interactions and intra-hemispheric rearrangement of the local neural activities in normal and pathological brain conditions.

We analyse the possible dynamical states emerging for two symmetrically pulse coupled populations of leaky integrate-and-fire neurons. In particular, we observe broken symmetry states in this set-up: namely, breathing chimeras, where one population is fully synchronized and the other is in a state of partial synchronization (PS) as well as generalized chimera states, where both populations are in PS, but with different levels of synchronization. Symmetric macroscopic states are also present, ranging from quasi-periodic motions, to collective chaos, from splay states to population anti-phase partial synchronization. We then investigate the influence of disorder, such as random link removal or noise, on the dynamics of collective solutions in this model. As a result, we observe that broken symmetry chimera-like states, with both populations partially synchronized, persist up to \(80 \%\) of broken links and up to noise amplitudes \({\simeq }8 \%\) of threshold-reset distance. Furthermore, the introduction of disorder on symmetric chaotic state has a constructive effect, namely to induce the emergence of chimera-like states at intermediate dilution or noise level.

The applications of nanotechnology in the field of neuroscience can be divided into two main strands: (i) applications in the field of basic research and (ii) applications in the clinical field. In the first area we deal with: (a) developing and applying nano-engineered materials to promote adhesion, growth and neuronal differentiation and to understand the neurobiological mechanisms underlying these processes; (b) fabricating nano-systems (for example, “nano-electrodes” implantable) for direct iteration, recording and stimulation of the neurons at the molecular level; (c) applying nano-structures and nanoscale resolution microscopy for advanced and better resolution imaging and diagnostics. In the clinical context, however, the primary goal is to limit or reverse the neurodegenerative processes. In this Lecture Note we present three different approaches at the crossing between basic research and application in clinical field. First, we report on the study of the effect of endogenous dipeptides in neurodegenerative diseases. Then we discuss some recent results in the field of the development of nano-engineered biocompatible materials (“scaffolds”) that might facilitate and accelerate neuronal growth, which represents one of the fundamental objectives of modern tissue engineering. As well, we describe the synthesis of biocompatible micro- and nano-systems that can transport small molecules, drugs, immune system or stem cells, through different routes of administration, a primary goal for the treatment of a wide family of neurological disorders, as well as brain tumors. Finally, we discuss the packaging of stimuli responsive composite systems for cell and cell surrounding environment monitoring, a new road now starting to be strongly pursued.

Tremendous efforts are made towards the development and realization of memristors for memory technology. Furthermore, memristor-based neuron and synapse models are considered in several investigations on neuromorphic systems. Some work is devoted to take advantage of peculiar nonlinear dynamics emerging in memristors to extend or improve the functionalities of state-of-the-art circuits and systems, both in the digital domain, where their use has been proposed in logic gate design, and in the analog domain, where volatile memristors with negative differential resistance effect, capable to amplify infinitesimal fluctuations of energy, have been adopted to design interesting transistor-less circuits. While in an increasingly number of cases, memristor models based on charge transport phenomena can be verified through measurements to a high degree of accuracy, the systematic design of circuits exploiting the rich dynamical behavior of memristors is still restricted to certain cases. For some memristors, even circuits with a few number of elements may exhibit different nonlinear phenomena, which can be only described by results obtained in numerical simulations. In this manuscript a nonlinear system theory-based approach will be introduced and discussed in detail. It is based on the determination of nonlinear characteristic functions allowing the characterization of circuit properties to a high degree of accuracy. These real-valued functions represent circuit elements or sub-circuits, and may be used in an automated circuit design approach. Results showing the performance of the proposed method will be given and discussed in this contribution.

The theoretical importance of memristor goes much beyond the field, i.e., circuit theory, in which its discovery originated. In fact, neuroscience and nonlinear science in general are also interested as memristors have been found to be adequate model of both synapses and Hodgkin–Huxley axons. In this work, we discuss the use of memristors as the unique nonlinear element of the cell of a cellular architecture reproducing several phenomena of interest for nonlinear science such as autowave propagation and Turing pattern formation. We illustrate the model and present numerical simulations showing how the same cell structure can account for these different dynamical behaviors when its parameters are varied.