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Linear stability in networks of pulse-coupled neurons

Simona Olmi
2014 Frontiers in Computational Neuroscience  
In this paper we discuss a general approach to linear stability of pulse-coupled neural networks for generic phase-response curves and post-synaptic response functions.  ...  In a first step towards the comprehension of neural activity, one should focus on the stability of the various dynamical states.  ...  Acknowledgments We thank David Angulo Garcia for the help in the use of symbolic algebra software.  ... 
doi:10.3389/fncom.2014.00008 pmid:24550817 pmcid:PMC3912513 fatcat:rthlyigxvfdsxgjovp4djo6pua

Stability criteria for splay states in networks of "generalized" neuronal models

Alessandro Torcini, Massimo Calamai, Antonio Politi
2009 BMC Neuroscience  
Splay states represent collective modes emerging in networks of fully coupled oscillators, they have been observed in Josephson junction arrays, laser systems, pulse-coupled neuronal networks [1] .  ...  The aim of the present analysis is to show that the SW part of the Floquet spectrum can be crucial for the stability of the splay states in fully pulse-coupled networks when generic neuronal models, different  ...  Splay states represent collective modes emerging in networks of fully coupled oscillators, they have been observed in Josephson junction arrays, laser systems, pulse-coupled neuronal networks [1] .  ... 
doi:10.1186/1471-2202-10-s1-p157 fatcat:pmsg22ol4vf3taefd7t565wgqi

Predicting n:1 locking in pulse coupled two-neuron networks using phase resetting theory

Fatma Gurel Kazanci, Selva K Maran, Astrid A Prinz, Carmen C Canavier
2008 BMC Neuroscience  
A stability analysis of the linearized map provides a single eigenvalue for the map, provided the second order resetting of all but the last input in a cycle is disregarded.  ...  Existence and stability criteria for harmonic locking modes were derived for two reciprocally pulse coupled oscillators based on their first and second order phase resetting curves (PRCs).  ... 
doi:10.1186/1471-2202-9-s1-p136 fatcat:crtc6kmmbjd5dovrnmo5ysjw4i

Splay states in networks of identical integrate-and-fire neurons

Jan R Engelbrecht, Bolun Chen, Renato Mirollo
2014 BMC Neuroscience  
We develop an analytic framework to investigate the stability of splay states in infinite networks of identical integrate-and-fire neurons coupled through synaptic pulses.  ...  More specifically we perform a linear stability analysis of the splay state probability distribution whose dynamics is governed by an appropriate Fokker Planck equation.  ...  , Canada. 26-31 July 2014 We develop an analytic framework to investigate the stability of splay states in infinite networks of identical integrate-and-fire neurons coupled through synaptic pulses.  ... 
doi:10.1186/1471-2202-15-s1-p91 pmcid:PMC4126587 fatcat:eb2vg3bavvhl5kqwuw5nhafz7y

Oscillations in an excitatory neuronal network with synaptic depression and adaptation

Zachary P Kilpatrick, Paul C Bressloff
2009 BMC Neuroscience  
Linear stability analysis of the space-clamped system indicates regions of parameter space wherein limit cycles exist.  ...  Previous modeling studies sought to explain such phenomena using non-locally coupled excitatory neuronal networks with local negative feedback [1].  ...  Linear stability analysis of the space-clamped system indicates regions of parameter space wherein limit cycles exist.  ... 
doi:10.1186/1471-2202-10-s1-p238 fatcat:ffiqnnzcsnholcwe7lnu3yg5sy

Stability of the splay state in pulse-coupled networks

Rüdiger Zillmer, Roberto Livi, Antonio Politi, Alessandro Torcini
2007 Physical Review E  
The stability of the dynamical states characterized by a uniform firing rate ( splay states) is analyzed in a network of globally coupled leaky integrate-and-fire neurons.  ...  This is done by reducing the set of differential equations to a map that is investigated in the limit of large network size.  ...  In this paper we also clarify the basic question whether a given network of pulse-coupled neurons exhibits a finite stability or if it is "marginally" stable.  ... 
doi:10.1103/physreve.76.046102 pmid:17995055 fatcat:nlq5uk3wkngrpjjmvc5muynsji

Phase response curves determine network activity of all to all networks of pulse coupled oscillators

Srisairam Achuthan, Carmen C Canavier
2008 BMC Neuroscience  
Stability for both cases was determined by constructing pulse coupled maps for an assumed firing order based on the PRCs by linearizing the coupled map.  ...  A pulse coupled simulator that does not assume any firing order, but merely takes an initial condition in terms of phases of the network oscillators and updates the phases on each cycle based on the PRC  ... 
doi:10.1186/1471-2202-9-s1-p135 fatcat:etismcfqere2tjrizbpfz5esa4

Globally attracting synchrony in a network of oscillators with all-to-all inhibitory pulse coupling

Carmen C. Canavier, Ruben A. Tikidji-Hamburyan
2017 Physical review. E  
We assume a network of all-to-all pulse-coupled oscillators in which the effect of a pulse is independent of the number of oscillators that simultaneously emit a pulse and the normalized delay (the phase  ...  We use phase resetting theory [23] [24] [25] and stability results based on event driven maps to prove that in a network of pulse coupled phase oscillators with a small conduction delay δ , inhibition  ...  In a subsequent paper [28] , they simulate identical all-to-all inhibitory pulse-coupled Hodgkin Huxley neurons with alpha functions with a delay, and get two to three clusters in networks of 50 neurons  ... 
doi:10.1103/physreve.95.032215 pmid:28415236 pmcid:PMC5568753 fatcat:gdfrj2rcvzgs7opkovf7zkc6am

Pulse coupled oscillators and the phase resetting curve

Carmen C. Canavier, Srisairam Achuthan
2010 Mathematical Biosciences  
PRCs can be used to predict phase locking in networks of pulse coupled oscillators.  ...  Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators.  ...  Similarly, as a function of the synaptic conductance strength in the four neuron network of Wang and Buzsaki model neurons Fig. 5 . 5 Existence and stability analysis for two bidirectionally pulse coupled  ... 
doi:10.1016/j.mbs.2010.05.001 pmid:20460132 pmcid:PMC3022482 fatcat:jr6hafyde5bylirz3oj26ocg4a

Page 689 of Mathematical Reviews Vol. , Issue 2001A [page]

2001 Mathematical Reviews  
The existence and stability of traveling waves and pulses in a one-dimensional network of integrate-and-fire (IF) neurons are studied.  ...  A general dynamical theory for networks of pulse-coupled integrate-and-fire (IF) neurons is derived that bridges the gap between weakly coupled phase oscillator models and strongly coupled firing rate  ... 

Non-Additive Coupling Enables Propagation of Synchronous Spiking Activity in Purely Random Networks

Raoul-Martin Memmesheimer, Marc Timme, Lyle J. Graham
2012 PLoS Computational Biology  
Here, we study spiking neural networks with non-additive dendritic interactions that were recently uncovered in single-neuron experiments.  ...  This study adds a novel perspective on the dynamics of networks with nonlinear interactions in general and presents a new viable mechanism for the occurrence of patterns of precisely timed spikes in recurrent  ...  Evolution of synchronous pulses in linearly (a) and nonlinearly (b) coupled networks.  ... 
doi:10.1371/journal.pcbi.1002384 pmid:22532791 pmcid:PMC3330086 fatcat:4xzoqjiwmfedncrjnkpafide6a

Splay States in Finite Pulse-Coupled Networks of Excitable Neurons

M. Dipoppa, M. Krupa, A. Torcini, B. S. Gutkin
2012 SIAM Journal on Applied Dynamical Systems  
The emergence and stability of splay states is studied in fully coupled finite networks of N excitable quadratic integrate-and-fire neurons, connected via synapses modeled as pulses of finite amplitude  ...  For M overlapping post synaptic potentials the linear stability analysis of the splay state should take in account, besides the actual values of the membrane potentials, also the firing times associated  ...  Linear stability for δ-pulses.  ... 
doi:10.1137/110859683 fatcat:mor5vpcca5dorhbor65rdi3h54

Stability of synchronous states in sparse neuronal networks [article]

Afifurrahman and Ekkehard Ullner and Antonio Politi
2020 arXiv   pre-print
The stability of synchronous states is analysed in the context of two populations of inhibitory and excitatory neurons, characterized by different pulse-widths.  ...  A detailed analysis, which includes also the study of finite-amplitude perturbations, is performed in the limit of narrow pulses, finding that the stability depends crucially on the relative pulse-width  ...  Here, however, we stick to pulse-coupled oscillators. The stability of the synchronised state of pulse-coupled phase oscillators has been first studied in the context of excitatory δ-pulses [15] .  ... 
arXiv:2002.00448v1 fatcat:52dverykwbeuhnh6rcbi2vdwoq

Collective dynamics in the presence of finite-width pulses [article]

Afifurrahman and Ekkehard Ullner and Antonio Politi
2021 arXiv   pre-print
In particular, we study two populations of identical excitatory and inhibitory neurons in a random network of phase oscillators coupled through exponential pulses with different widths.  ...  We investigate the effect of a finite pulse-width on the dynamics of balanced neuronal networks.  ...  In particular, we study two populations of identical excitatory and inhibitory neurons in a random network of phase oscillators coupled through exponential pulses with different widths.  ... 
arXiv:2102.03438v2 fatcat:bsaudza7h5dphmxj3s7bsofysa

Stability of synchronous states in sparse neuronal networks [article]

Afifurrahman, Ekkehard Ullner, Antonio Politi
2020 bioRxiv   pre-print
The stability of synchronous states is analysed in the context of two populations of inhibitory and excitatory neurons, characterized by different pulse-widths.  ...  A detailed analysis, which includes also the study of finite-amplitude perturbations, is performed in the limit of narrow pulses, finding that the stability depends crucially on the relative pulse-width  ...  Here, however, we stick to pulse-coupled oscillators. The stability of the synchronised state of pulse-coupled phase oscillators has been first studied in the context of excitatory δ-pulses [15] .  ... 
doi:10.1101/2020.02.02.931048 fatcat:idqgwcc77ngbdhlatv43ubwlty
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