the influence of limit cycle topology on the phase resetting curve understanding the phenomenology of phase resetting is an essential step toward developing a formalism for the analysis of circuits composed of bursting neurons that receive multiple and sometimes overlapping inputs if we are to use phase resetting methods to analyze these circuits we can either generate phase resetting curves prcs for all possible inputs and combinations of inputs or we can develop an understanding of how to construct prcs for arbitrary perturbations of a given neuron the latter strategy is the goal of this study we present a geometrical derivation of phase resetting of neural limit cycle oscillators in response to short current pulses a geometrical phase is defined as the distance traveled along the limit cycle in the appropriate phase space the perturbations in current are treated as displacements in the direction corresponding to membrane voltage we show that for type i oscillators the direction of a perturbation in current is nearly tangent to the limit cycle hence the projection of the displacement in voltage onto the limit cycle is sufficient to give the geometrical phase resetting in order to obtain the phase resetting in terms of elapsed time or temporal phase a mapping between geometrical and temporal phase is obtained empirically and used to make the conversion this mapping is shown to be an invariant of the dynamics perturbations in current applied to type ii oscillators produce significant normal displacements from the limit cycle so the difference in angular velocity at displaced points compared to the angular velocity on the limit cycle must be taken into account empirical attempts to correct for differences in angular velocity amplitude versus phase effects in terms of a circular coordinate system during relaxation back to the limit cycle achieved some success in the construction of phase resetting curves for type ii model oscillators the ultimate goal of this work is the extension of these techniques to biological circuits comprising type ii neural oscillators which appear frequently in identified central pattern generating circuits

networks of theta neurons with time varying excitability macroscopic chaos multistability and final state uncertainty abstract using recently developed analytical techniques we study the macroscopic dynamics of a large heterogeneous network of theta neurons in which the neurons excitability parameter varies in time we demonstrate that such periodic variation can lead to the emergence of macroscopic chaos multistability and final state uncertainty in the collective behavior of the network finite size network effects and rudimentary control via an accessible macroscopic network parameter is also investigated

chaotic pattern alternations can reproduce properties of dominance durations in multistable perception we propose a pulse neural network that exhibits chaotic pattern alternations among stored patterns as a model of multistable perception which is reflected in phenomena such as binocular rivalry and perceptual ambiguity when we regard the mixed state of patterns as a part of each pattern the durations of the retrieved pattern obey unimodal distributions we confirmed that no chaotic properties are observed in the time series of durations consistent with the findings of previous psychological studies moreover it is shown that our model also reproduces two properties of multistable perception that characterize the relationship between the contrast of inputs and the durations

coupling regularizes individual units in noisy populations the regularity of a noisy system can modulate in various ways it is well known that coupling in a population can lower the variability of the entire network the collective activity is more regular here we show that diffusive reciprocal coupling of two simple ornstein uhlenbeck o u processes can regularize the individual even when it is coupled to a noisier process in cellular networks the regularity of individual cells is important when a select few play a significant role the regularizing effect of coupling surprisingly applies also to general nonlinear noisy oscillators however unlike with the o u process coupling induced regularity is robust to different kinds of coupling with two coupled noisy oscillators we derive an asymptotic formula assuming weak noise and coupling for the variance of the period i e spike times that accurately captures this effect moreover we find that reciprocal coupling can regularize the individual period of higher dimensional oscillators such as the morris lecar and brusselator models even when coupled to noisier oscillators coupling can have a counterintuitive and beneficial effect on noisy systems these results have implications for the role of connectivity with noisy oscillators and the modulation of variability of individual oscillators

slowly coupled oscillators phase dynamics and synchronization in this paper we extendthe results of frankel andkiemel siam j appl math 5 3 1993 pp 1436 1446 to a network of slowly coupledoscillators first we use malkin s theorem to derive a canonical phase model that describes synchronization properties of a slowly coupled network then we illustrate the result using slowly coupledoscillators 1 near and ronov hopf bifurcations 2 near saddle node on invariant circle bifurcations and 3 near relaxation oscillations we compare andcontrast synchronization properties of slowly andweakly coupledoscillators

variability in noise driven integrator neurons neural variability in the presence of noise has been studied mainly in resonator neurons such as hodgkinhuxley or fitzhugh nagumo models here we investigate this variability for integrator neurons whose excitability is due to a saddle node bifurcation of the rest state instead of a hopf bifurcation using simple theoretical expressions for the interspike times distributions we obtain coefficients of variation in good agreement with numerical calculations in realistic neuron models the main features of this coefficient as a function of noise depend on the refractory period and on the presence of bistability the bistability is responsible for the existence of two different time scales in the spiking behavior giving an antiresonance effect

time scales of spike train correlation for neural oscillators with common drive we examine the effect of the phase resetting curve on the transfer of correlated input signals into correlated output spikes in a class of neural models receiving noisy superthreshold stimulation we use linear response theory to approximate the spike correlation coefficient in terms of moments of the associated exit time problem and contrast the results for type i vs type ii models and across the different time scales over which spike correlations can be assessed we find that on long time scales type i oscillators transfer correlations much more efficiently than type ii oscillators on short time scales this trend reverses with the relative efficiency switching at a time scale that depends on the mean and standard deviation of input currents this switch occurs over time scales that could be exploited by downstream circuits

nonlinear coupling near a degenerate hopf bautin bifurcation a nonlinearly coupled system of bistable fixed point and limit cycle differential equations is analyzed the nonlinear equations arise from the first several terms in the normal form expansion near a bautin bifurcation existence and stability of in phase and out of phase periodic solutions to a pair of identical systems are explored existence uniqueness and stability of traveling wave solutions from a stable rest state to a stable periodic solution are proved for the associated evolution convolution equation numerical simulations suggest some interesting patterns in regimes where waves no longer exist the results are shown to hold for a nonreduced conductance based model

improving the precision of noisy oscillators abstract we consider how the period of an oscillator is affected by white noise with special attention given to the cases of additive noise and parameter fluctuations our treatment is based upon the concepts of isochrons which extend the notion of the phase of a stable periodic orbit to the basin of attraction of the periodic orbit and phase response curves which can be used to understand the geometry of isochrons near the periodic orbit this includes a derivation of the leading order effect of noise on the statistics of an oscillator s period several examples are considered in detail which illustrate the use and validity of the theory and demonstrate how to improve a noisy oscillator s precision by appropriately tuning system parameters or operating away from a bifurcation point it is also shown that appropriately timed impulsive kicks can give further improvements to oscillator precision

multipulse phase resetting curves in this paper we introduce and study systematically in terms of phase response curves the effect of dual pulse excitation on the dynamics of an autonomous oscillator specifically we test the deviations from linear summation of phase advances resulting from two small perturbations we analytically derive a correction term which generally appears for oscillators whose intrinsic dimensionality is g 1 the nonlinear correction term is found to be proportional to the square of the perturbation we demonstrate this effect in the stuart landau model and in various higher dimensional neuronal models this deviation from the superposition principle needs to be taken into account in studies of networks of pulse coupled oscillators further this deviation could be used in the verification of oscillator models via a dual pulse excitation

white noise induced spiral waves and multiple spatial coherence resonances in a neuronal network with type i excitability abstract white noise induced pattern formation is studied in a network composed of morris lecar neuronal models with type i excitability and with initial values higher than that of the resting potential the appearance and disappearance of spiral waves as well as the transitions between spiral wave patterns with different kinds of complexity characterized by the normalized spatial autocorrelation function enable changes in the order of the network so as to exhibit a scenario with two or more locally maximal peaks as can be clearly seen in the signal to noise ratio curves as the noise intensity is adjusted from small to large in a wide range a possible physical mechanism of the multiple resonances based on the dynamics of type i excitability and initial values is provided the potential biological significance of the noise induced spiral waves is discussed

alexander invariants of complex hyperplane arrangements leta be an arrangement of n complex hyperplanes the funda mental group of the complement of a is determined by a braid monodromy homomorphism fi fs pn using the gassner representation of the pure braid group we flnd an explicit presentation for the alexander invariant ofa from this presentation we obtain combinatorial lower bounds for the ranks of the chen groups ofa we also provide a combinatorial criterion for when these lower bounds are attained

control and synchronization of neuron ensembles synchronization of oscillations is a phenomenon prevalent in natural social and engineering systems controlling synchronization of oscillating systems is motivated by a wide range of applications from surgical treatment of neurological diseases to the design of neurocomputers in this paper we study the control of an ensemble of uncoupled neuron oscillators described by phase models we examine controllability of such a neuron ensemble for various phase models and furthermore study the related optimal control problems in particular by employing pontryagin s maximum principle we analytically derive optimal controls for spiking single and two neuron systems and analyze the applicability of the latter to an ensemble system finally we present a robust computational method for optimal control of spiking neurons based on pseudospectral approximations the methodology developed here is universal to the control of general nonlinear phase oscillators

type i membranes phase resetting curves and synchrony type i membrane oscillators such as the connor model connor et al 1977 and the morris lecar model morris and lecar 1981 admit very low frequency oscillations near the critical applied current hansel et al 1995 have numerically shown that synchrony is difficult to achieve with these models and that the phase resetting curve is strictly positive we use singular perturbation methods and averaging to show that this is a general property of type i membrane models we show in a limited sense that so called type ii resetting occurs with models that obtain rhythmicity via a hopf bifurcation we also show the differences between synapses that act rapidly and those that act slowly and derive a canonical form for the phase interactions

the shape of phase resetting curves in oscillators with a saddle node on an invariant circle bifurcation we introduce a simple two dimensional model that extends the poincare oscillator so that the attracting limit cycle undergoes a saddle node bifurcation on an invariant circle snic for certain parameter values arbitrarily close to this bifurcation the phase resetting curve prc continuously depends on parameters where its shape can be not only primarily positive or primarily negative but also nearly sinusoidal this example system shows that one must be careful inferring anything about the bifurcation structure of the oscillator from the shape of its prc

rewiring induced chaos in pulse coupled neural networks the dependence of the dynamics of pulse coupled neural networks on random rewiring of excitatory and inhibitory connections is examined when both excitatory and inhibitory connections are rewired periodic synchronization emerges with a hopf like bifurcation and a subsequent period doubling bifurcation chaotic synchronization is also observed when only excitatory connections are rewired periodic synchronization emerges with a saddle node like bifurcation and chaotic synchronization is also observed this result suggests that randomness in the system does not necessarily contaminate the system and sometimes it even introduces rich dynamics to the system such as chaos

phase resetting of collective rhythm in ensembles of oscillators phase resetting curves characterize the way a system with a collective periodic behavior responds to perturbations we consider globally coupled ensembles of sakaguchi kuramoto oscillators and use the ott antonsen theory of ensemble evolution to derive the analytical phase resetting equations we show the final phase reset value to be composed of two parts an immediate phase reset directly caused by the perturbation and the dynamical phase reset resulting from the relaxation of the perturbed system back to its dynamical equilibrium analytical semianalytical and numerical approximations of the final phase resetting curve are constructed we support our findings with extensive numerical evidence involving identical and nonidentical oscillators the validity of our theory is discussed in the context of large ensembles approximating the thermodynamic limit

multivariable alexander invariants of hypersurface complements we start with a discussion on alexander invariants and then prove some general results concerning the divisibility of the alexander polynomials and the supports of the alexander modules via artin s vanishing theorem for perverse sheaves we conclude with explicit computations of twisted cohomology following an idea already exploited in the hyperplane arrangement case which combines the degeneration of the hodge to de rham spectral sequence with the purity of some cohomology groups

class ii neurons display a higher degree of stochastic synchronization than class i neurons we describe the relationship between the shape of the phase resetting curve prc and the degree of stochastic synchronization observed between a pair of uncoupled general oscillators receiving partially correlated poisson inputs in addition to inputs from independent sources we use perturbation methods to derive an expression relating the shape of the prc to the probability density function pdf of the phase difference between the oscillators we compute various measures of the degree of synchrony and cross correlation from the pdf s and use the same to compare and contrast differently shaped prcs with respect to their ability to undergo stochastic synchronization since the shape of the prc depends on underlying dynamical details of the oscillator system we utilize the results obtained from the analysis of general oscillator systems to study specific models of neuronal oscillators it is shown that the degree of stochastic synchronization is controlled both by the firing rate of the neuron and the membership of the prc type i or type ii it is also shown that the circular variance for the integrate and fire neuron and the generalized order parameter for a hippocampal interneuron model have a nonlinear relationship to the input correlation

predictability of critical transitions critical transitions in multistable systems have been discussed as models for a variety of phenomena ranging from the extinctions of species to socioeconomic changes and climate transitions between ice ages and warm ages from bifurcation theory we can expect certain critical transitions to be preceded by a decreased recovery from external perturbations the consequences of this critical slowing down have been observed as an increase in variance and autocorrelation prior to the transition however especially in the presence of noise it is not clear whether these changes in observation variables are statistically relevant such that they could be used as indicators for critical transitions in this contribution we investigate the predictability of critical transitions in conceptual models we study the quadratic integrate and fire model and the van der pol model under the influence of external noise we focus especially on the statistical analysis of the success of predictions and the overall predictability of the system the performance of different indicator variables turns out to be dependent on the specific model under study and the conditions of accessing it furthermore we study the influence of the magnitude of transitions on the predictive performance

synchronization of weakly coupled canard oscillators synchronization has been studied extensively in the context of weakly coupled oscillators using the so called phase response curve prc which measures how a change of the phase of an oscillator is affected by a small perturbation this approach was based upon the work of malkin and it has been extended to relaxation oscillators namely synchronization conditions were established under the weak coupling assumption leading to a criterion for the existence of synchronous solutions of weakly coupled relaxation oscillators previous analysis relies on the fact that the slow nullcline does not intersect the fast nullcline near one of its fold points where canard solutions can arise in the present study we use numerical continuation techniques to solve the adjoint equations and we show that synchronization properties of canard cycles are different than those of classical relaxation cycles in particular we highlight a new special role of the maximal canard in separating two distinct synchronization regimes the hopf regime and the relaxation regime phase plane analysis of slow fast oscillators undergoing a canard explosion provides an explanation for this change of synchronization properties across the maximal canard

local linear approximation of the jacobian matrix better captures phase resetting of neural limit cycle oscillators one effect of any external perturbations such as presynaptic inputs received by limit cycle oscillators when they are part of larger neural networks is a transient change in their firing rate or phase resetting a brief external perturbation moves the figurative point outside the limit cycle a geometric perturbation that we mapped into a transient change in the firing rate or a temporal phase resetting in order to gain a better qualitative understanding of the link between the geometry of the limit cycle and the phase resetting curve prc we used a moving reference frame with one axis tangent and the others normal to the limit cycle we found that the stability coefficients associated with the unperturbed limit cycle provided good quantitative predictions of both the tangent and the normal geometric displacements induced by external perturbations a geometric to temporal mapping allowed us to correctly predict the prc while preserving the intuitive nature of this geometric approach

on the phase reduction and response dynamics of neural oscillator populations weundertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli recalling and extending results from the literature we compute phase response curves prcs valid near bifurcations to periodic firing for hindmarsh rose hodgkin huxley fitzhugh nagumo and morris lecar models encompassing the four generic codimension one bifurcations phase density equations are then used to analyze the role of the bifurcation and the resulting prc in responses to stimuli in particular we explore the interplay among stimulus duration baseline firing frequency and population level response patterns we interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions

the geometry of neuronal recruitment we address the question of whether or not a periodic train of excitatory synaptic inputs recruits an excitable cell such that it fires repeatedly or does not recruit a cell such that it fails to fire possibly after some transient in particular we study the scenarios of one or two inputs per period in the latter case the degree of synchrony of the inputs is a crucial factor in recruitment we establish rigorous geometric conditions that pinpoint the transition between recruitment and non recruitment as the degree of synchrony between input pairs or other input parameters is varied these conditions can be used to determine whether a particular temporal relation between periodic input pairs leads to recruitment or not and to prove in certain parameter regimes that recruitment can only occur when the inputs are sufficiently closely synchronized the concepts in this paper are derived for both the integrate and fire neuron and the theta neuron models in the former the location in phase space of the unique fixed point of a relevant two dimensional map determines firing while in the latter it is the existence or lack of existence of a fixed point of the map which does so these results are discussed in the context of recruitment of cells into localized activity patterns

stationary bumps in networks of spiking neurons we examine the existence and stability of spatially localized bumps of neuronal activity in a network of spiking neurons bumps have been proposed in mechanisms of visual orientation tuning the rat head direction system and working memory we show that a bump solution can exist in a spiking network provided the neurons fire asynchronously within the bump we consider a parameter regime where the bump solution is bistable with an all off state and can be initiated with a transient excitatory stimulus we show that the activity profile matches that of a corresponding population rate model the bump in a spiking network can lose stability through partial synchronization to either a traveling wave or the all off state this can occur if the synaptic timescale is too fast through a dynamical effect or if a transient excitatory pulse is applied to the network a bump can thus be activated and deactivated with excitatory inputs that may have physiological relevance

identification and continuity of the distributions of burst length and interspike intervals in the stochastic morris lecar neuron using the morris lecar model neuron with a type ii parameter set and k channel noise we investigate the interspike interval distribution as increasing levels of applied current drive the model through a subcritical hopf bifurcation our goal is to provide a quantitative description of the distributions associated with spiking as a function of applied current the model generates bursty spiking behavior with sequences of random numbers of spikes bursts separated by interburst intervals of random length this kind of spiking behavior is found in many places in the nervous system most notably perhaps in stuttering inhibitory interneurons in cortex here we show several practical and inviting aspects of this model combining analysis of the stochastic dynamics of the model with estimation based on simulations we show that the parameter of the exponential tail of the interspike interval distribution is in fact continuous over the entire range of plausible applied current regardless of the bifurcations in the phase portrait of the model further we show that the spike sequence length apparently studied for the first time here has a geometric distribution whose associated parameter is continuous as a function of applied current over the entire input range hence this model is applicable over a much wider range of applied current than has been thought

asynchronous states and the emergence of synchrony in large networks of interacting excitatory and inhibitory neurons we investigate theoretically the conditions for the emergence of synchronous activity in large networks consisting of two populations of extensively connected neurons one excitatory and one inhibitory the neurons are modeled with quadratic integrate and fire dynamics which provide a very good approximation for the subthreshold behavior of a large class of neurons in addition to their synaptic recurrent inputs the neurons receive a tonic external input that varies from neuron to neuron because of its relative simplicity this model can be studied analytically we investigate the stability of the asynchronous state as of the network with given average firing rates of the two populations first we show that the as can remain stable even if the synaptic couplings are strong then we investigate the conditions under which this state can be destabilized we show that this can happen in four generic ways the first is a saddle node bifurcation which leads to another state with different average firing rates this bifurcation which occurs for strong enough recurrent excitation does not correspond to the emergence of synchrony in contrast in the three other instability mechanisms hopf bifurcations which correspond to the emergence of oscillatory synchronous activity occur we show that these mechanisms can be differentiated by the firing patterns they generate and their dependence on the mutual interactions of the inhibitory neurons and cross talk between the two populations we also show that besides these codimension 1 bifurcations the system can display several codimension 2 bifurcations takens bogdanov gavrielov guckenheimer and double hopf bifurcations

topology and geometry of cohomology jump loci we elucidate the key role played by formality in the theory of characteristic and resonance varieties we define relative characteristic and resonance varieties v k and r k related to twisted group cohomology with coefficients of arbitrary rank we show that the germs at the origin of v k and r k are analytically isomorphic if the group is 1 formal in particular the tangent cone to v k at 1 equals r k these new obstructions to 1 formality lead to a striking rationality property of the usual resonance varieties a detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth complex quasi projective algebraic variety this sheds new light on a classical problem of j p serre applications to arrangements configuration spaces coproducts of groups and artin groups are given

on synchrony of weakly coupled neurons at low firing rate the dynamics of a pair of weakly interacting conductance based neurons firing at low frequency is investigated in the framework of the phase reduction method the stability of the antiphase and the in phase locked state is studied it is found that for a large class of conductance based models the antiphase state is stable resp unstable for excitatory resp inhibitory interactions if the synaptic time constant is above a critical value sc which scales as log when goes to zero

design of charge balanced time optimal stimuli for spiking neuron oscillators in this letter we investigate the fundamental limits on how the interspike time of a neuron oscillator can be perturbed by the application of a bounded external control input a current stimulus with zero net electric charge accumulation we use phase models to study the dynamics of neurons and derive charge balanced controls that achieve the minimum and maximum interspike times for a given bound on the control amplitude our derivation is valid for any arbitrary shape of the phase response curve and for any value of the given control amplitude bound in addition we characterize the change in the structures of the charge balanced time optimal controls with the allowable control amplitude we demonstrate the applicability of the derived optimal control laws by applying them to mathematically ideal and experimentally observed neuron phase models including the widely studied hodgkin huxley phase model and by verifying them with the corresponding original full state space models this work addresses a fundamental problem in the field of neural control and provides a theoretical investigation to the optimal control of oscillatory systems

index reduction in differential algebraic equations using dummy derivatives a new index reduction algorithm for daes is developed in the usual manner parts of the dae are differentiated analytically and appended to the original system for each additional equation a derivative is selected to be replaced by a new algebraic variable called a dummy derivative the resulting augmented system is at most index 1 but no longer overdetermined the dummy derivatives are not subject to discretization their purpose is to annihilate part of the dynamics in the dae leaving only what corresponds to the dynamics of a state space form no constraint stabilization is necessary in the subsequent numerical treatment numerical tests indicate that the method yields results with an accuracy comparable to that obtained for the corresponding state space ode

the signature method for daes arising in the modeling of electrical circuits abstract we consider the signature method method for the structural analysis of differential algebraic equations daes that arise in the modeling and simulation of electrical circuits different formulations of the set of model equations are considered we show that for some formulations the structural approach may fail for certain circuit topologies while other formulations are better suited for a structural analysis in particular we show that for the branch oriented model equations the signature method always succeeds with a structural index that corresponds to the differentiation index of the system the results are illustrated by a number of examples

two cluster bifurcations in systems of globally pulse coupled oscillators abstract for a system of globally pulse coupled phase oscillators we derive conditions for stability of the completely synchronous state and all stationary two cluster states and explain how the different states are naturally connected via bifurcations the coupling is modeled using the phase response curve prc which measures the sensitivity of each oscillator s phase to perturbations for large systems with a prc which is zero at the spiking threshold we are able to find the parameter regions where multiple stable two cluster states coexist and illustrate this by an example in addition we explain how a locally unstable one cluster state may form an attractor together with its homoclinic connections this leads to the phenomenon of intermittent asymptotic synchronization with abating beats away from the perfect synchrony

interplay between excitability type and distributions of neuronal connectivity determines neuronal network synchronization while the interplay between neuronal excitability properties and global properties of network topology is known to affect network propensity for synchronization it is not clear how detailed characteristics of these properties affect spatiotemporal pattern formation here we study mixed networks composed of neurons having type i and or type ii phase response curves with varying distributions of local and random connections and show that not only average network properties but also the connectivity distribution statistics significantly affect network synchrony namely we study networks with fixed networkwide properties but vary the number of random connections that nodes project we show that varying node excitability type i vs type ii influences network synchrony most dramatically for systems with long tailed distributions of the number of random connections per node this indicates that a cluster of even a few highly rewired cells with a high propensity for synchronization can alter the degree of synchrony in the network as a whole we show this effect generally on a network of coupled kuramoto oscillators and investigate the impact of this effect more thoroughly in pulse coupled networks of biophysical neurons

firing rate response of linear and nonlinear integrate and fire neurons to modulated current based and conductance based synaptic drive integrate and fire models are mainstays of the study of single neuron response properties and emergent states of recurrent networks of spiking neurons they also provide an analytical base for perturbative approaches that treat important biological details such as synaptic filtering synaptic conductance increase and voltage activated currents steady state firing rates of both linear and nonlinear integrate and fire models receiving fluctuating synaptic drive can be calculated from the time independent fokker planck equation the dynamic firing rate response is less easy to extract even at the first order level of a weak modulation of the model parameters but is an important determinant of neuronal response and network stability for the linear integrate and fire model the response to modulations of current based synaptic drive can be written in terms of hypergeometric functions for the nonlinear exponential and quadratic models no such analytical forms for the response are available here it is demonstrated that a rather simple numerical method can be used to obtain the steady state and dynamic response for both linear and nonlinear models to parameter modulation in the presence of current based or conductance based synaptic fluctuations to complement the full numerical solution generalized analytical forms for the high frequency response are provided a special case is also identified time constant modulation for which the response to an arbitrarily strong modulation can be calculated exactly

multispikes and synchronization in a large neural network with temporal delays coherent rhythms in the gamma frequency range are ubiquitous in the nervous system and thought to be important in a variety of cognitive activities such rhythms are known to be able to synchronize with millisecond precision across distances with significant conduction delay it is mysterious how this can operate in a setting in which cells receive many inputs over a range of time here we analyze a version of mechanism previously proposed that the synchronization in the ca1 region of the hippocampus depends on the firing of doublets by the interneurons using a network of local circuits that are arranged in a possibly disordered lattice we determine the conditions on parameters for existence and stability of synchronous solutions in which the inhibitory interneurons fire single spikes doublets or triplets per cycle we show that the synchronous solution is only marginally stable if the interneurons fire singlets if they fire doublets the synchronous state is asymptotically stable in a larger subset of parameter space than if they fire triplets an unexpected finding is that a small amount of disorder in the lattice structure enlarges the parameter regime in which the doublet solution is stable synaptic noise reduces the regime in which the doublet configuration is stable but only weakly

characteristic varieties of hypersurface complements abstract we give divisibility results for the global characteristic varieties of hypersurface complements expressed in terms of the local characteristic varieties at points along one of the irreducible components of the hypersurface as an application we recast old and obtain new finiteness and divisibility results for the classical infinite cyclic alexander modules of complex hypersurface complements moreover for the special case of hyperplane arrangements we translate our divisibility results for characteristic varieties in terms of the corresponding resonance varieties

qualitative changes in phase response curve and synchronization at the saddle node loop bifurcation prominent changes in neuronal dynamics have previously been attributed to a specific switch in onset bifurcation the bogdanov takens bt point this study unveils another relevant and so far underestimated transition point the saddle node loop bifurcation which can be reached by several parameters including capacitance leak conductance and temperature this bifurcation turns out to induce even more drastic changes in synchronization than the bt transition this result arises from a direct effect of the saddle node loop bifurcation on the limit cycle and hence spike dynamics in contrast the bt bifurcation exerts its immediate influence upon the subthreshold dynamics and hence only indirectly relates to spiking we specifically demonstrate that the saddle node loop bifurcation i ubiquitously occurs in planar neuron models with a saddle node on invariant cycle onset bifurcation and ii results in a symmetry breaking of the system s phase response curve the latter entails an increase in synchronization range in pulse coupled oscillators such as neurons the derived bifurcation structure is of interest in any system for which a relaxation limit is admissible such as josephson junctions and chemical oscillators

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

effects of dendritic load on the firing frequency of oscillating neurons we study the effects of passive dendritic properties on the dynamics of neuronal oscillators we find that the addition of a passive dendrite can sometimes have counterintuitive effects on firing frequency specifically the addition of a hyperpolarized passive dendritic load can either increase decrease or have negligible effects on firing frequency we use the theory of weak coupling to derive phase equations for ball and stick model neurons and two compartment model neurons we then develop a framework for understanding how the addition of passive dendrites modulates the frequency of neuronal oscillators we show that the average value of the neuronal oscillator s phase response curves measures the sensitivity of the neuron s firing rate to the dendritic load including whether the addition of the dendrite causes an increase or decrease in firing frequency we interpret this finding in terms of to the slope of the neuronal oscillator s frequency applied current curve we also show that equivalent results exist for constant and noisy point source input to the dendrite we note that the results are not specific to neurons but are applicable to any oscillator subject to a passive load

measurement of infinitesimal phase response curves from noisy real neurons we sought to measure infinitesimal phase response curves iprcs from rat hippocampal ca1 pyramidal neurons it is difficult to measure iprcs from noisy neurons because of the dilemma that either the linearity or the signal to noise ratio of responses to external perturbations must be sacrificed to overcome this difficulty we used an iprc measurement model formulated as the langevin phase equation lpe to extract iprcs in the bayesian scheme we then simultaneously verified the effectiveness of the measurement model and the reliability of the estimated iprcs by demonstrating that lpes with the estimated iprcs could predict the stochastic behaviors of the same neurons whose iprcs had been measured when they were perturbed by periodic stimulus currents our results suggest that the lpe is an effective model for real oscillating neurons and that many theoretical frameworks based on it may be applicable to real nerve systems

consistent spectral predictors for dynamic causal models of steady state responses dynamic causal modelling dcm for steady state responses ssr is a framework for inferring the mechanisms that underlie observed electrophysiological spectra using biologically plausible generative models of neuronal dynamics in this paper we examine the dynamic repertoires of nonlinear conductance based neural population models and propose a generative model of their power spectra our model comprises an ensemble of interconnected excitatory and inhibitory cells where synaptic currents are mediated by fast glutamatergic and gabaergic receptors and slower voltage gated nmda receptors we explore two formulations of how hidden neuronal states depolarisation and conductances interact through their mean and variance mean field model or through their mean alone neural mass model both rest on a nonlinear fokker planck description of population dynamics which can exhibit bifurcations phase transitions we first characterise these phase transitions numerically by varying critical model parameters we elicit both fixed points and quasiperiodic dynamics that reproduce the spectral characteristics 2 100 hz of real electrophysiological data we then introduce a predictor of spectral activity using centre manifold theory and linear stability analysis this predictor is based on sampling the system s jacobian over the orbits of hidden neuronal states this predictor behaves consistently and smoothly in the region of phase transitions which permits the use of gradient descent methods for model inversion we demonstrate this by inverting generative models dcms of ssrs using simulated data that entails phase transitions

interaction function of oscillating coupled neurons large scale simulations of electrically coupled neuronal oscillators often employ the phase coupled oscillator paradigm to understand and predict network behavior we study the nature of the interaction between such coupled oscillators using weakly coupled oscillator theory by employing piecewise linear approximations for phase response curves and voltage time courses and parametrizing their shapes we compute the interaction function for all such possible shapes and express it in terms of discrete fourier modes we find that reasonably good approximation is achieved with four fourier modes that comprise of both sine and cosine terms

regular traveling waves in a one dimensional network of theta neurons this paper focuses on regular traveling waves or waves that propagate smoothly in a one dimensional network of theta neurons we show that when coupling strength is sufficiently large there exist two traveling wave solutions for this network moreover the cells in the network spike more than one time after joining the wave that is some form of synaptic depression or other adaptive mechanism must exist to attain single spike traveling waves we also present results about how the behavior of solutions to this network depends on the maximal synaptic coupling strength and the wave speed this is relevant to consideration of uniqueness and to bounding the range of coupling strengths for which waves exist

evaluation of entrainment of a nonlinear neural oscillator to white noise the lyapunov exponent for a one dimensional neural oscillator model the theta neuron is computed for white noise forcing using the steady state solution to the associated fokker planck equation the latter is mildly singular due to the nature of the multiplicative input in agreement with previous results with similar models the exponent is negative for all forcing amplitudes but here it is shown to be small relative to that for periodic drive in a range of forcing strengths thus the synchronization of an ensemble of independent neurons receiving common but random input can be slow moreover this implies that aperiodic input may be suboptimal in some contexts for preserving the reliability of fine spike timing a potentially important component of the neural code

optimal entrainment with smooth pulse and square signals in weakly forced nonlinear oscillators abstract a physical limit of entrainability of nonlinear oscillators is considered for an external weak signal forcing this limit of entrainability is characterized by the optimization problem maximizing the width of the arnold tongue the frequency locking range versus forcing magnitude under certain practical constraints here we show a solution to this optimization problem thanks to a direct link to holder s inequality this solution defines an ideal forcing realizing the entrainment limit and as the result a fundamental limit of entrainment is clarified as follows for 1 1 entrainment we obtain i a construction of the global optimal forcing and a condition for its uniqueness in l p space with p 1 and ii a construction of the global optimal pulse like forcings in l 1 space and for m n entrainment m n some informations about the non existence of the ideal forcing iii in addition we establish definite algorithms for obtaining the global optimal forcings for 1 p and these pulse like forcings for p 1 these theoretical findings are verified by systematic extensive numerical calculations and simulations

synchronization in networks of excitatory and inhibitory neurons with sparse random connectivity in model networks of e cells and i cells excitatory and inhibitory neurons respectively synchronous rhythmic spiking often comes about from the interplay between the two cell groups the e cells synchronize the i cells and vice versa under ideal conditions homogeneity in relevant network parameters and all to all connectivity for instance this mechanism can yield perfect synchronization we find that approximate imperfect synchronization is possible even with very sparse random connectivity the crucial quantity is the expected number of inputs per cell as long as it is large enough more precisely as long as the variance of the total number of synaptic inputs per cell is small enough tight synchronization is possible the desynchronizing effect of random connectivity can be reduced by strengthening the e i synapses more surprising it cannot be reduced by strengthening the i e synapses however the decay time constant of inhibition plays an important role faster decay yields tighter synchrony in particular in models in which the inhibitory synapses are assumed to be instantaneous the effects of sparse random connectivity cannot be seen

generalization of coupled spiking models and effects of the width of an action potential on synchronization phenomena the integrate and fire if neuron model and the piecewise linear version of fitzhugh nagumo pl neuron model with a time scale parameter mu are frequently being used in the study of synchronization phenomena although the two models are regarded to be different type we show a certain equivalence between them by deriving the coupled if model of an improved version with a firing duration from the recovery variable coupled system of the pl model under taking the limit of mu 0 without a loss of any coupling properties in the coupled if model with the duration time the synchronization behavior of a pair of neurons with excitatory or inhibitory synaptic coupling can be systematically explored in terms of three parameters in the model synaptic strength decaying relaxation rate of the synaptic coupling and the parameter exhibiting the firing duration we find some irregularly synchronous behavior with or without constant firing order alternations we show that the duration of an impulse plays an important role in synchronization phenomena

increasing inhibitory input increases neuronal firing rate why and when diffusion process cases increasing inhibitory input to single neuronal models such as the fitzhugh nagumo model and the hodgkin huxley model can sometimes increase their firing rates a phenomenon which we term inhibition boosted firing ibf here we consider neuronal models with diffusion approximation inputs i e they share the identical first and second order statistics of the corresponding poisson process inputs using the integrate and fire model and the if fhn model we explore theoretically how and when ibf can happen for both models it is shown that there is a critical input frequency at which the efferent firing rate is identical when the neuron receives purely excitatory inputs or exactly balanced inhibitory and excitatory inputs when the input frequency is lower than the critical frequency ibf occurs

phase desynchronization a new approach and theory using pulse based interaction in this paper we present a novel approach for achieving phase desynchronization in a pulse coupled oscillator network ensuring phase desynchronization is a difficult problem and existing results are constrained to a completely interconnected network and a fixed number of nodes our approach is more robust than previous approaches removing the constraint of a fixed number of nodes the removal of this constraint is significant because it allows the network to receive and drop nodes freely without any change to the phase update strategy also to our knowledge our approach is the first to prove the convergence to the desynchronized state for a topology that is more general than the all to all topology more specifically our approach is applicable to any circulant symmetric network topology including the circulant symmetric ring topology rigorous mathematical proofs are provided to support the result that any circulant symmetric network with ordered phases under our proposed algorithm will converge to uniform phase desynchronization simulation results are also presented to demonstrate the algorithm s performance

optimal design of minimum power stimuli for phase models of neuron oscillators in this paper we study optimal control problems of spiking neurons whose dynamics are described by a phase model we design minimum power current stimuli controls that lead to targeted spiking times in particular we consider bounded control amplitude and characterize the range of possible spiking times determined by the bound which can be chosen sufficiently small within the range where the phase model is valid we show that for a given bound the corresponding feasible spiking times are optimally achieved by piecewise continuous controls we present analytic expressions with numerical simulations of the minimum power stimuli for several phase models we demonstrate the applicability of our method with an experimentally determined phase response curve

on admissible rank one local systems abstract a rank one local system l on a smooth complex algebraic variety m is 1 admissible if the dimension of the first cohomology group h 1 m l can be computed from the cohomology algebra h m c in degrees 2 under the assumption that m is 1 formal we show that all local systems except finitely many on a non translated irreducible component w of the first characteristic variety v 1 m are 1 admissible see proposition 3 1 the same result holds for local systems on a translated component w but now h m c should be replaced by h m 0 c where m 0 is a zariski open subset obtained from m by deleting some hypersurfaces determined by the translated component w see theorem 4 3 one consequence of this result is that the local systems l where the dimension of h 1 m l jumps along a given positive dimensional component of the characteristic variety v 1 m have finite order see theorem 4 7 using this we show in corollary 4 9 that dim h 1 m l dim h 1 m l 1 for any rank one local system l on a smooth complex algebraic variety m

phase reduction theory based treatment of extended delayed feedback control algorithm in the presence of a small time delay mismatch the delayed feedback control dfc methods are noninvasive which means that the control signal vanishes if the delay time is adjusted to be equal to the period of a target unstable periodic orbit upo if the delay time differs slightly from the upo period a nonvanishing periodic control signal is observed we derive an analytical expression for this period for a general class of multiple input multiple output systems controlled by an extended dfc algorithm our approach is based on the phase reduction theory adapted to systems with time delay the analytical results are supported by numerical simulations of the controlled rossler system

emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback interaction via pulses is common in many natural systems especially neuronal in this article we study one of the simplest possible systems with pulse interaction a phase oscillator with delayed pulsatile feedback when the oscillator reaches a specific state it emits a pulse which returns after propagating through a delay line the impact of an incoming pulse is described by the oscillator s phase reset curve prc in such a system we discover an unexpected phenomenon for a sufficiently steep slope of the prc a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value the number of such critical multipliers increases linearly with the delay and thus may be arbitrary large this bifurcation is accompanied by the emergence of numerous jittering regimes with nonequal interspike intervals isis each of these regimes corresponds to a periodic solution of the system with a period roughly proportional to the delay the number of different jittering solutions emerging at the bifurcation point increases exponentially with the delay we describe the combinatorial mechanism that underlies the emergence of such a variety of solutions in particular we show how a periodic solution exhibiting several distinct isis can imply the existence of multiple other solutions obtained by rearranging of these isis we show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback

phase locking in weakly heterogeneous neuronal networks abstract we examine analytically the existence and stability of phase locked states in a weakly heterogeneous neuronal network we consider a model of n neurons with all to all synaptic coupling where the heterogeneity is in the intrinsic firing frequency of the individual neurons we consider both inhibitory and excitatory coupling we derive the conditions under which stable phase locking is possible in homogeneous networks many different periodic phase locked states are possible their stability depends on the dynamics of the neuron and the coupling for weak heterogeneity the phase locked states are perturbed from the homogeneous states and can remain stable if their homogeneous counterparts are stable for enough heterogeneity phase locked solutions either lose stability or are destroyed completely we analyze the possible states the network can take when phase locking is broken

predicting the existence and stability of phase locked mode in neural networks using generalized phase resetting curve we used the phase resetting method to study a biologically relevant three neuron network in which one neuron receives multiple inputs per cycle for this purpose we first generalized the concept of phase resetting to accommodate multiple inputs per cycle we explicitly showed how analytical conditions for the existence and the stability of phase locked modes are derived in particular we solved newly derived recursive maps using as an example a biologically relevant driving driven neural network with a dynamic feedback loop we applied the generalized phase resetting definition to predict the relative phase and the stability of a phase locked mode in open loop setup we also compared the predicted phase locked mode against numerical simulations of the fully connected network

continuum of weakly coupled oscillatory mckean neurons the mckean model of a neuron possesses a one dimensional fast voltage like variable and a slow recovery variable a recent geometric analysis of the singularly perturbed system has allowed an explicit construction of its phase response curve s coombes 2001 phase locking in networks of synaptically coupled mckean relaxation oscillators physica d vol 160 173 188 here we use tools from coupled oscillator theory to study weakly coupled networks of mckean neurons using numerical techniques we show that the mckean system has traveling wave phase locked solutions consistent with that of a network of more biophysically detailed hodgkin huxley neurons

dynamics of membrane excitability determine interspike interval variability a link between spike generation mechanisms and cortical spike train statistics we propose a biophysical mechanism for the high interspike interval variability observed in cortical spike trains the key lies in the nonlinear dynamics of cortical spike generation which are consistent with type i membranes where saddle node dynamics underlie excitability rinzel ermentrout 1989 we present a canonical model for type i membranes the neuron the neuron is a phase model whose dynamics reflect salient features of type i membranes this model generates spike trains with coefficient of variation cv above 0 6 when brought to firing by noisy inputs this happens because the timing of spikes for a type i excitable cell is exquisitely sensitive to the amplitude of the suprathreshold stimulus pulses a noisy input current giving random amplitude kicks to the cell evokes highly irregular firing across a wide range of firing rates an intrinsically oscillating cell gives regular spike trains we corroborate the results with simulations of the morris lecar m l neural model with random synaptic inputs type i m l yields high cvs when this model is modified to have type ii dynamics periodicity arises via a hopf bifurcation however it gives regular spike trains cv below 0 3 our results suggest that the high cv values such as those observed in cortical spike trains are an intrinsic characteristic of type i membranes driven to firing by random inputs in contrast neural oscillators or neurons exhibiting type ii excitability should produce regular spike trains

effect of phase response curve skewness on synchronization of electrically coupled neuronal oscillators we investigate why electrically coupled neuronal oscillators synchronize or fail to synchronize using the theory of weakly coupled oscillators stability of synchrony and antisynchrony is predicted analytically and verified using numerical bifurcation diagrams the shape of the phase response curve prc the shape of the voltage time course and the frequency of spiking are freely varied to map out regions of parameter spaces that hold stable solutions we find that type 1 and type 2 prcs can hold both synchronous and antisynchronous solutions but the shape of the prc and the voltage determine the extent of their stability this is achieved by introducing a five piecewise linear model to the prc and a three piecewise linear model to the voltage time course and then analyzing the resultant eigenvalue equations that determine the stability of the phase locked solutions a single time parameter defines the skewness of the prc and another single time parameter defines the spike width and frequency our approach gives a comprehensive picture of the relation of the prc shape voltage time course and stability of the resultant synchronous and antisynchronous solutions

phase equations for relaxation oscillators we use the malkin theorem to derive phase equations for networks of weakly connected relaxation oscillators we find an explicit formula for the connection functions when the oscillators have one dimensional slow variables the functions are discontinuous in the relaxation limit mu rightarrow 0 which provides a simple alternative illustration to the major conclusion of the fast threshold modulation ftm theory by somers and kopell biological cybernetics 68 1993 pp 393 407 that synchronization of relaxation oscillators has properties that are quite different from those of smooth nonrelaxation oscillators we use bonhoeffer van der pol relaxation oscillators to illustrate the theory numerically

differential algebraic equations of index 1 may have an arbitrarily high structural index it is shown that the structural index of a linear differential algebraic equation dae with constant coefficients and with index 1 may be arbitrarily high contrary to a previous result from the literature this demonstrates that pantelides algorithm applied to daes of index 1 may perform an arbitrarily high number of iterations and differentiations

the index of general nonlinear daes in the last few years there has been considerable research on differential algebraic equations daes f t y y 0 where f y is identically singular much of the mathematical effort has focused on computing a solution that is assumed to exist more recently there has been some discussion of solvability of daes there has historically been some imprecision in the use of the two key concepts of solvability and index for daes the index is also important in control and systems theory but with different terminology the consideration of increasingly complex nonlinear daes makes a clear and correct development necessary this paper will try to clarify several points concerning the index after establishing some new and more precise terminology that we need some inaccuracies in the literature will be corrected the two types of indices most frequently used the differentiation index and the perturbation index are defined with respect to solutions of unperturbed problems examples are given to show that these indices can be very different for the same problem we define new maximum indices which are the maxima of earlier indices in a neighborhood of the solution over a set of perturbations and show that these indices are simply related to each other these indices are also related to an index defined in terms of jacobians