Noise, transient dynamics, and the generation of realistic interspike interval variation in square-wave burster neurons

First return maps of interspike intervals for biological neurons that generate repetitive bursts of impulses can display stereotyped structures (neuronal signatures). Such structures have been linked to the possibility of multicoding and multifunctionality in neural networks that produce and control rhythmical motor patterns. In some cases, isolating the neurons from their synaptic network revealsirregular, complex signatures that have been regarded as evidence of intrinsic, chaotic behavior. We show that incorporation of dynamical noise into minimal neuron models of square-wave bursting (either conductance-based or abstract) produces signatures akin to those observed in biological examples, without the need for fine-tuning of parameters or ad hoc constructions for inducing chaotic activity. The form of the stochastic term is not strongly constrained, and can approximate several possible sources of noise, e.g. random channel gating or synaptic bombardment. The cornerstone of this signature generation mechanism is the rich, transient, but deterministic dynamics inherent in the square-wave (saddle-node/homoclinic) mode of neuronal bursting. We show that noise causes the dynamics to populate a complex transient scaffolding or skeleton in state space, even for models that (without added noise) generate only periodic activity (whether in bursting or tonic spiking mode).Comment: REVTeX4-1, 18 pages, 9 figure

Unconventional criticality, scaling breakdown, and diverse universality classes in the Wilson-Cowan model of neural dynamics

M.A.M. acknowledges the Spanish Ministry and Agencia Estatal de investigación (AEI) through Project of Ref. PID2020-113681GB-I00, financed by MICIN/AEI/10.13039/501100011033 and FEDER “A way to make Europe,” as well as the Consejería de Conocimiento, Investigación Universidad, Junta de Andalucía, and European Regional Development Fund, Project No. P20-00173 for financial support. H.C.P. acknowledges CAPES (PrInT Grant No. 88887.581360/2020-00) and is grateful for the hospitality of the Statistical Physics group at the Instituto Interuniversitario Carlos I de Física Teórica y Computacional at the University of Granada during her six-month stay, during which part of this work was developed. M.C. acknowledges support by CNPq (Grants No. 425329/2018-6 and No. 308703/2022-7), CAPES (Grant No. PROEX 23038.003069/2022-87), and FACEPE (Grant No. APQ-0642-1.05/18). This article was produced as part of the activities of Programa Institucional de Internacionalização (PrInt). We are also very thankful to R. Corral, S. di Santo, V. Buendia, J. Pretel, and I. L. D. Pinto for valuable discussions and comments on previous versions of the manuscript.The Wilson-Cowan model constitutes a paradigmatic approach to understanding the collective dynamics of networks of excitatory and inhibitory units. It has been profusely used in the literature to analyze the possible phases of neural networks at a mean-field level, e.g., assuming large fully connected networks. Moreover, its stochastic counterpart allows one to study fluctuation-induced phenomena, such as avalanches. Here we revisit the stochastic Wilson-Cowan model paying special attention to the possible phase transitions between quiescent and active phases. We unveil eight possible types of such transitions, including continuous ones with scaling behavior belonging to known universality classes—such as directed percolation and tricritical directed percolation—as well as six distinct ones. In particular, we show that under some special circumstances, at a so-called “Hopf tricritical directed percolation” transition, rather unconventional behavior is observed, including the emergence of scaling breakdown. Other transitions are discontinuous and show different types of anomalies in scaling and/or exhibit mixed features of continuous and discontinuous transitions. These results broaden our knowledge of the possible types of critical behavior in networks of excitatory and inhibitory units and are, thus, of relevance to understanding avalanche dynamics in actual neuronal recordings. From a more general perspective, these results help extend the theory of nonequilibrium phase transitions into quiescent or absorbing states.Consejería de ConocimientoInstituto Interuniversitario Carlos I de Física Teórica y Computacional at the University of GranadaInvestigación Universidad, Junta de AndalucíaSpanish MinistryCoordenação de Aperfeiçoamento de Pessoal de Nível Superior 88887.581360/2020-00 CAPESConselho Nacional de Desenvolvimento Científico e Tecnológico 23038.003069/2022-87, 308703/2022-7, 425329/2018-6 CNPqFundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco APQ-0642-1.05/18 FACEPEEuropean Regional Development Fund P20-00173 ERDFAgencia Estatal de Investigación MICIN/AEI/10.13039/501100011033, PID2020-113681GB-I00 AE

Temporal structure of bursting patterns as representation of input history

From Eighteenth Annual Computational Neuroscience Meeting: CNS*2009 Berlin, Germany. 18–23 July 2009.This work is supported by Brazilian agencies Fapesp and CAPES, CNPq, and Spanish grants MEC PHB2007-0013TA, BFU2006-07902/BFI, TIN 2007- 65989 and CAM S-SEM-0255-2006

High prevalence of multistability of rest states and bursting in a database of a model neuron.

Flexibility in neuronal circuits has its roots in the dynamical richness of their neurons. Depending on their membrane properties single neurons can produce a plethora of activity regimes including silence, spiking and bursting. What is less appreciated is that these regimes can coexist with each other so that a transient stimulus can cause persistent change in the activity of a given neuron. Such multistability of the neuronal dynamics has been shown in a variety of neurons under different modulatory conditions. It can play either a functional role or present a substrate for dynamical diseases. We considered a database of an isolated leech heart interneuron model that can display silent, tonic spiking and bursting regimes. We analyzed only the cases of endogenous bursters producing functional half-center oscillators (HCOs). Using a one parameter (the leak conductance (g(leak)) bifurcation analysis, we extended the database to include silent regimes (stationary states) and systematically classified cases for the coexistence of silent and bursting regimes. We showed that different cases could exhibit two stable depolarized stationary states and two hyperpolarized stationary states in addition to various spiking and bursting regimes. We analyzed all cases of endogenous bursters and found that 18% of the cases were multistable, exhibiting coexistences of stationary states and bursting. Moreover, 91% of the cases exhibited multistability in some range of g(leak). We also explored HCOs built of multistable neuron cases with coexisting stationary states and a bursting regime. In 96% of cases analyzed, the HCOs resumed normal alternating bursting after one of the neurons was reset to a stationary state, proving themselves robust against this perturbation

One parameter bifurcation diagrams of stationary states of the robust burster cases from the leech heart interneuron database.

<p>The bifurcation parameter is the conductance of the leak current, . The solid and dashed intervals of a bifurcation curve denote stable and unstable stationary state branches, respectively. Labeled points in panels B–F indicate bifurcations of stationary states: LP and AH stand for the fold and Andronov-Hopf bifurcations, respectively. The navy blue bars indicate the ranges of values supporting attracting bursting regimes, while the colored bars above them denote ranges supporting the coexistence of distinct attracting regimes. The color code of the bars is described in the key and is consistent between figures (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002930#pcbi-1002930-g002" target="_blank">Figures 2</a>,<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002930#pcbi-1002930-g004" target="_blank">4</a>,<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002930#pcbi-1002930-g006" target="_blank">6</a>). Panels D, and F are magnifications of the diagrams around the range supporting bursting in panels C and E, respectively. The vertical navy blue lines in panels B, D, and F and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002930#pcbi-1002930-g004" target="_blank">Figure 4B</a> indicate the original value for the case in the database. A: A stylized bifurcation diagram illustrating the naming convention for the intervals on the branches where stationary states are stable: <i>hyp1</i>, <i>hyp2</i>, <i>dep1</i>, and <i>dep2</i>. B: Magnification of the bifurcation diagram for case #288298 that displays a stable interval <i>hyp2</i> on the middle branch delimited by two Andronov-Hopf bifurcations, AH2 and AH3. The green bar denotes a range supporting bursting and the <i>hyp2</i> stationary state. C and D: In the case #1292494 there are three stable stationary state intervals. C: The red brown bar indicates a range of values supporting the coexistence of <i>dep1</i> and <i>dep2</i> stationary states. The Andronov-Hopf bifurcation (AH2) and the fold bifurcation LP3 determine the <i>dep1</i> interval. The fold bifurcation (LP4) defines the right border of the <i>dep2</i> interval. The left border of the bistability range is 0 nS. C and D: The magenta bar marks coexistence of <i>dep1</i> and bursting. The bar is limited by the range supporting bursting on the left side and by Andronov-Hopf bifurcation AH2 on the right side. The cyan bar indicates coexistence of <i>hyp1</i> and bursting between the Andronov-Hopf Bifurcation (AH1) and the border where bursting disappears. E and F: A case where the original value in the database falls into a range of bistability of bursting and <i>hyp1</i> stationary state (cyan bar). The bistability range is defined by the Andronov-Hopf bifurcation (AH1) and the right border where bursting disappears.</p

A classification scheme for the cases with multistability.

<p>The bifurcation diagrams describe stationary states branches as the conductance of the leak current is varied and have the ranges supporting bursting marked. There are 10 multistability arrangements classified here: (A) bursting and the <i>hyp1</i>; (B) bursting with <i>hyp2</i>; (C) a range supporting <i>hyp1</i> and bursting along with a range supporting <i>dep1</i> and <i>dep2</i>; (D) a range of bistability of <i>hyp1</i> and bursting along with a range of bistability of <i>dep1</i> and bursting; (E) one range of coexisting <i>hyp1</i> and <i>dep1</i> and one range of coexisting <i>dep1</i> and bursting; (F) three ranges of coexistence of exactly two regimes: (1) <i>dep1</i> and <i>dep2</i>, (2) <i>dep1</i> and bursting, and (3) <i>hyp1</i> and bursting; (G) one range of tristability along with a range of coexistence of <i>hyp1</i> and <i>dep1</i> and a range of coexistence of <i>dep1</i> and bursting; (H) one range of tristability along with a range of bistability of <i>hyp1</i> and bursting and a range of bistability of <i>dep1</i> and bursting; (I) Four ranges of multistability: a range of tristability along with a range of bistability of <i>hyp1</i> and <i>dep1</i>, a range of bistability of <i>dep1</i> and bursting, and a range of bistability of <i>dep1</i> and <i>dep2</i>; (J) two ranges of bistability of <i>hyp1</i> and <i>dep1</i> in addition to a range supporting tristability. The solid (dotted) curves represent stable (unstable) branches. The navy blue bar underlies the range supporting bursting activity. The other colored bars mark different types of multistability as indicated in a key.</p

A histogram of prevalence of the multistability of the silent and bursting regimes.

<p>Prevalence of multistability was calculated as the percentage of the whole range of values supporting bursting that supports multistability of the stationary states and robust bursting.</p