67 research outputs found
Nonlinear Dynamics, Synchronisation and Chaos in Coupled FHN Cardiac and Neural Cells
Physiological systems are amongst the most challenging systems to investigate from a
mathematically based approach. The eld of mathematical biology is a relatively recent
one when compared to physics. In this thesis I present an introduction to the physiological
aspects needed to gain access to both cardiac and neural systems for a researcher trained
in a mathematically based discipline. By using techniques from nonlinear dynamical
systems theory I show a number of results that have implications for both neural and
cardiac cells. Examining a reduced model of an excitable biological oscillator I show how
rich the dynamical behaviour of such systems can be when coupled together. Quantifying
the dynamics of coupled cells in terms of synchronisation measures is treated at length.
Most notably it is shown that for cells that themselves cannot admit chaotic solutions,
communication between cells be it through electrical coupling or synaptic like coupling,
can lead to the emergence of chaotic behaviour. I also show that in the presence of
emergent chaos one nds great variability in intervals of activity between the constituent
cells. This implies that chaos in both cardiac and neural systems can be a direct result
of interactions between the constituent cells rather than intrinsic to the cells themselves.
Furthermore the ubiquity of chaotic solutions in the coupled systems may be a means of
information production and signaling in neural systems
Multiscale Computations on Neural Networks: From the Individual Neuron Interactions to the Macroscopic-Level Analysis
We show how the Equation-Free approach for multi-scale computations can be
exploited to systematically study the dynamics of neural interactions on a
random regular connected graph under a pairwise representation perspective.
Using an individual-based microscopic simulator as a black box coarse-grained
timestepper and with the aid of simulated annealing we compute the
coarse-grained equilibrium bifurcation diagram and analyze the stability of the
stationary states sidestepping the necessity of obtaining explicit closures at
the macroscopic level. We also exploit the scheme to perform a rare-events
analysis by estimating an effective Fokker-Planck describing the evolving
probability density function of the corresponding coarse-grained observables
Mechanisms explaining transitions between tonic and phasic firing in neuronal populations as predicted by a low dimensional firing rate model
Several firing patterns experimentally observed in neural populations have
been successfully correlated to animal behavior. Population bursting, hereby
regarded as a period of high firing rate followed by a period of quiescence, is
typically observed in groups of neurons during behavior. Biophysical
membrane-potential models of single cell bursting involve at least three
equations. Extending such models to study the collective behavior of neural
populations involves thousands of equations and can be very expensive
computationally. For this reason, low dimensional population models that
capture biophysical aspects of networks are needed.
\noindent The present paper uses a firing-rate model to study mechanisms that
trigger and stop transitions between tonic and phasic population firing. These
mechanisms are captured through a two-dimensional system, which can potentially
be extended to include interactions between different areas of the nervous
system with a small number of equations. The typical behavior of midbrain
dopaminergic neurons in the rodent is used as an example to illustrate and
interpret our results.
\noindent The model presented here can be used as a building block to study
interactions between networks of neurons. This theoretical approach may help
contextualize and understand the factors involved in regulating burst firing in
populations and how it may modulate distinct aspects of behavior.Comment: 25 pages (including references and appendices); 12 figures uploaded
as separate file
Neuronal oscillations: from single-unit activity to emergent dynamics and back
L’objectiu principal d’aquesta tesi és avançar en la comprensió del processament d’informació en xarxes neuronals en presència d’oscil lacions subumbrals. La majoria de neurones propaguen la seva activitat elèctrica a través de sinapsis quÃmiques que són activades, exclusivament, quan el corrent elèctric que les travessa supera un cert llindar. És per aquest motiu que les descà rregues rà pides i intenses produïdes al soma neuronal, els anomenats potencials d’acció, són considerades la unitat bà sica d’informació neuronal, és a dir, el senyal mÃnim i necessari per a iniciar la comunicació entre dues neurones. El codi neuronal és entès, doncs, com un llenguatge binari que expressa qualsevol missatge (estÃmul sensorial, memòries, etc.) en un tren de potencials d’acció. Tanmateix, cap funció cognitiva rau en la dinà mica d’una única neurona. Circuits de milers de neurones connectades entre sà donen lloc a determinats ritmes, palesos en registres d’activitat colectiva com els electroencefalogrames (EEG) o els potencials de camp local (LFP). Si els potencials d’acció de cada cèl lula, desencadenats per fluctuacions estocà stiques de les corrents sinà ptiques, no assolissin un cert grau de sincronia, no apareixeria aquesta periodicitat a nivell de xarxa.
Per tal de poder entendre si aquests ritmes intervenen en el codi neuronal hem estudiat tres situacions. Primer, en el CapÃtol 2, hem mostrat com una
cadena oberta de neurones amb un potencial de membrana intrÃnsecament oscil latori filtra un senyal periòdic arribant per un dels extrems. La resposta
de cada neurona (pulsar o no pulsar) depèn de la seva fase, de forma que cada una d’elles rep un missatge filtrat per la precedent. A més, cada potencial
d’acció presinà ptic provoca un canvi de fase en la neurona postsinà ptica que depèn de la seva posició en l’espai de fases. Els perÃodes d’entrada capaços de sincronitzar les oscil lacions subumbrals són aquells que mantenen la fase d’arribada dels potencials d’acció fixa al llarg de la cadena. Per tal de què el missatge arribi intacte a la darrera neurona cal, a més a més, que aquesta fase permeti la descà rrega del voltatge transmembrana.
En segon cas, hem estudiat una xarxa neuronal amb connexions tant a veïns propers com de llarg abast, on les oscil lacions subumbrals emergeixen de
l’activitat col lectiva reflectida en els corrents sinà ptics (o equivalentment en el LFP). Les neurones inhibidores aporten un ritme a l’excitabilitat de la
xarxa, és a dir, que els episodis en què la inhibició és baixa, la probabilitat d’una descà rrega global de la població neuronal és alta. En el CapÃtol 3
mostrem com aquest ritme implica l’aparició d’una bretxa en la freqüència de descà rrega de les neurones: o bé polsen espaiadament en el temps o bé
en rà fegues d’elevada intensitat. La fase del LFP determina l’estat de la xarxa neuronal codificant l’activitat de la població: els mÃnims indiquen la
descà rrega simultà nia de moltes neurones que, ocasionalment, han superat el llindar d’excitabilitat degut a un decreixement global de la inhibició, mentre
que els mà xims indiquen la coexistència de rà fegues en diferents punts de la xarxa degut a decreixements locals de la inhibició en estats globals d’excitació. Aquesta dinà mica és possible grà cies al domini de la inhibició sobre l’excitació. En el CapÃtol 4 considerem acoblament entre dues xarxes neuronals per tal d’estudiar la interacció entre ritmes diferents. Les oscil lacions indiquen recurrència en la sincronització de l’activitat col lectiva, de manera que durant aquestes finestres temporals una població optimitza el seu impacte en una xarxa diana. Quan el ritme de la població receptora i el de l’emissora difereixen significativament, l’eficiència en la comunicació decreix, ja que les fases de mà xima resposta de cada senyal LFP no mantenen una diferència constant entre elles.
Finalment, en el CapÃtol 5 hem estudiat com les oscil lacions col lectives pròpies de l’estat de son donen lloc al fenomen de coherència estocà stica.
Per a una intensitat òptima del soroll, modulat per l’excitabilitat de la xarxa, el LFP assoleix una regularitat mà xima donant lloc a un perÃode refractari de
la població neuronal.
En resum, aquesta Tesi mostra escenaris d’interacció entre els potencials d’acció, caracterÃstics de la dinà mica de neurones individuals, i les oscil lacions
subumbrals, fruit de l’acoblament entre les cèl lules i ubiqües en la dinà mica de poblacions neuronals. Els resultats obtinguts aporten funcionalitat a aquests
ritmes emergents, agents sincronitzadors i moduladors de les descà rregues neuronals i reguladors de la comunicació entre xarxes neuronals.The main objective of this thesis is to better understand information processing in neuronal networks in the presence of subthreshold oscillations. Most neurons propagate their electrical activity via chemical synapses, which are only activated when the electric current that passes through them surpasses a certain threshold. Therefore, fast and intense discharges produced at the neuronal soma (the action potentials or spikes) are considered the basic unit of neuronal information. The neuronal code is understood, then, as a binary language that expresses any message (sensory stimulus, memories, etc.) in a train of action potentials. Circuits of thousands of interconnected neurons give rise to certain rhythms, revealed in collective activity measures such as electroencephalograms (EEG) and local field potentials (LFP). Synchronization of action potentials of each cell, triggered by stochastic fluctuations of the synaptic currents, cause this periodicity at the network level.To understand whether these rhythms are involved in the neuronal code we studied three situations. First, in Chapter 2, we showed how an open chain of neurons with an intrinsically oscillatory membrane potential filters a periodic signal coming from one of its ends. The response of each neuron (to spike or not) depends on its phase, so that each cell receives a message filtered by the preceding one. Each presynaptic action potential causes a phase change in the postsynaptic neuron, which depends on its position in the phase space. Those incoming periods that are able to synchronize the subthreshold oscillations, keep the phase of arrival of action potentials fixed along the chain. The original message reaches intact the last neuron provided that this phase allows the discharge of the transmembrane voltage.I the second case, we studied a neuronal network with connections to both long range and close neighbors, in which the subthreshold oscillations emerge from the collective activity apparent in the synaptic currents. The inhibitory neurons provide a rhythm to the excitability of the network. When inhibition is low, the likelihood of a global discharge of the neuronal population is high. In Chapter 3 we show how this rhythm causes a gap in the discharge frequency of neurons: either they pulse single spikes or they fire bursts of high intensity. The LFP phase determines the state of the neuronal network, coding the activity of the population: its minima indicate the simultaneous discharge of many neurons, while its maxima indicate the coexistence of bursts due to local decreases of inhibition at global states of excitation. In Chapter 4 we consider coupling between two neural networks in order to study the interaction between different rhythms. The oscillations indicate recurrence in the synchronization of collective activity, so that during these time windows a population optimizes its impact on a target network. When the rhythm of the emitter and receiver population differ significantly, the communication efficiency decreases as the phases of maximum response of each LFP signal do not maintain a constant difference between them.Finally, in Chapter 5 we studied how oscillations typical of the collective sleep state give rise to stochastic coherence. For an optimal noise intensity, modulated by the excitability of the network, the LFP reaches a maximal regularity leading to a refractory period of the neuronal population.In summary, this Thesis shows scenarios of interaction between action potentials, characteristics of the dynamics of individual neurons, and the subthreshold oscillations, outcome of the coupling between the cells and ubiquitous in the dynamics of neuronal populations . The results obtained provide functionality to these emerging rhythms, triggers of synchronization and modulator agents of the neuronal discharges and regulators of the communication between neuronal networks
Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models
The presence of gap junction coupling among neurons of the central nervous systems has been appreciated for some time now. In recent years there has been an upsurge of interest from the mathematical community in understanding the contribution of these direct electrical connections between cells to large-scale brain rhythms. Here we analyze a class of exactly soluble single neuron models, capable of producing realistic action potential shapes, that can be used as the basis for understanding dynamics at the network level. This work focuses on planar piece-wise linear models that can mimic the firing response of several different cell types. Under constant current injection the periodic response and phase response curve (PRC) is calculated in closed form. A simple formula for the stability of a periodic orbit is found using Floquet theory. From the calculated PRC and the periodic orbit a phase interaction function is constructed that allows the investigation of phase-locked network states using the theory of weakly coupled oscillators. For large networks with global gap junction connectivity we develop a theory of strong coupling instabilities of the homogeneous, synchronous and splay state. For a piece-wise linear caricature of the Morris-Lecar model, with oscillations arising from a homoclinic bifurcation, we show that large amplitude oscillations in the mean membrane potential are organized around such unstable orbits
Synchronization of electrically coupled resonate-and-fire neurons
Electrical coupling between neurons is broadly present across brain areas and
is typically assumed to synchronize network activity. However, intrinsic
properties of the coupled cells can complicate this simple picture. Many cell
types with strong electrical coupling have been shown to exhibit resonant
properties, and the subthreshold fluctuations arising from resonance are
transmitted through electrical synapses in addition to action potentials. Using
the theory of weakly coupled oscillators, we explore the effect of both
subthreshold and spike-mediated coupling on synchrony in small networks of
electrically coupled resonate-and-fire neurons, a hybrid neuron model with
linear subthreshold dynamics and discrete post-spike reset. We calculate the
phase response curve using an extension of the adjoint method that accounts for
the discontinuity in the dynamics. We find that both spikes and resonant
subthreshold fluctuations can jointly promote synchronization. The subthreshold
contribution is strongest when the voltage exhibits a significant post-spike
elevation in voltage, or plateau. Additionally, we show that the geometry of
trajectories approaching the spiking threshold causes a "reset-induced shear"
effect that can oppose synchrony in the presence of network asymmetry, despite
having no effect on the phase-locking of symmetrically coupled pairs
Roles of gap junctions in neuronal networks
This dissertation studies the roles of gap junctions in the dynamics of neuronal networks in three distinct problems. First, we study the circumstances under which a network of excitable cells coupled by gap junctions exhibits sustained activity. We investigate how network connectivity and refractory length affect the sustainment of activity in an abstract network. Second, we build a mathematical model for gap junctionally coupled cables to understand the voltage response along the cables as a function of cable diameter. For the coupled cables, as cable diameter increases, the electrotonic distance decreases, which cause the voltage to attenuate less, but the input of the second cable decreases, which allows the voltage of the second cable to attenuate more. Thus we show that there exists an optimal diameter for which the voltage amplitude in the second cable is maximized. Third, we investigate the dynamics of two gap-junctionally coupled theta neurons. A single theta neuron model is a canonical form of Type I neural oscillator that yields a very low frequency oscillation. The coupled system also yields a very low frequency oscillation in the sense that the ratio of two cells\u27 spiking frequencies obtains the values from a very small number. Thus the network exhibits several types of solutions including stable suppressed and 1 N spiking solutions. Using phase plane analysis and Denjoy\u27s Theorem, we show the existence of these solutions and investigate some of their properties
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