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The Geometry of Spontaneous Spiking in Neuronal Networks

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Abstract

The mathematical theory of pattern formation in electrically coupled networks of excitable neurons forced by small noise is presented in this work. Using the Freidlin–Wentzell large-deviation theory for randomly perturbed dynamical systems and the elements of the algebraic graph theory, we identify and analyze the main regimes in the network dynamics in terms of the key control parameters: excitability, coupling strength, and network topology. The analysis reveals the geometry of spontaneous dynamics in electrically coupled network. Specifically, we show that the location of the minima of a certain continuous function on the surface of the unit n-cube encodes the most likely activity patterns generated by the network. By studying how the minima of this function evolve under the variation of the coupling strength, we describe the principal transformations in the network dynamics. The minimization problem is also used for the quantitative description of the main dynamical regimes and transitions between them. In particular, for the weak and strong coupling regimes, we present asymptotic formulas for the network activity rate as a function of the coupling strength and the degree of the network. The variational analysis is complemented by the stability analysis of the synchronous state in the strong coupling regime. The stability estimates reveal the contribution of the network connectivity and the properties of the cycle subspace associated with the graph of the network to its synchronization properties. This work is motivated by the experimental and modeling studies of the ensemble of neurons in the Locus Coeruleus, a nucleus in the brainstem involved in the regulation of cognitive performance and behavior.

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Notes

  1. Positive parameter b in the definition of D (cf. (4.12)) is used to exclude the possibility of exit from D through ∂D∖∂ + D.

  2. Note that l i≠0,i∈[n], as long as v i is not an isolated node of G. In particular, if G is connected then \(l^{i}\neq0\;\forall i\in[n]\).

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Acknowledgement

This work was partly supported by the NSF Award DMS 1109367 (to GM).

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Authors and Affiliations

  1. Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA, 19104, USA

    Georgi S. Medvedev

  2. Bank of America, 1200 N. King St., Wilmington, DE, 19801, USA

    Svitlana Zhuravytska

Authors
  1. Georgi S. Medvedev
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  2. Svitlana Zhuravytska
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Correspondence to Georgi S. Medvedev.

Additional information

Communicated by P. Newton.

Appendix: The Parameter Values Used in the Biophysical Model (2.1) and (2.2)

Appendix: The Parameter Values Used in the Biophysical Model (2.1) and (2.2)

To emphasize that the results of this study do not rely on any specific features of the LC neuron model, in our numerical experiments we used the Morris–Lecar model, a common Type I biophysical model of an excitable cell (Rinzel and Ermentrout 1989). This model is based on the Hodgkin–Huxley paradigm. The function on the right hand side of the voltage equation (2.1), I ion=I Ca+I K+I l, models the combined effect of the calcium and sodium currents, I Ca, the potassium current, I K, and a small leak current, I l,

The constants E Ca, E K, and E l stand for the reversal potentials and g Ca, g K, and g l denote the maximal conductances of the corresponding ionic currents. The activation of the calcium and potassium channels are modeled using the steady-state functions

$$m_\infty(v)=0.5 \biggl(1+\tanh \biggl(\frac{v-\nu_1}{\nu_2} \biggr) \biggr)\quad\mbox{and}\quad n_\infty(v)=0.5 \biggl(1+\tanh \biggl(\frac{v-\nu_3}{\nu_4} \biggr) \biggr),$$

and the voltage-dependent time constant

$$\tau(v)= \biggl(\cosh \biggl(\frac{v-\nu_3}{2 \nu_4} \biggr) \biggr)^{-1}.$$

The parameter values are summarized in the Table 1.

Table 1 Parameter values
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Medvedev, G.S., Zhuravytska, S. The Geometry of Spontaneous Spiking in Neuronal Networks. J Nonlinear Sci 22, 689–725 (2012). https://doi.org/10.1007/s00332-012-9125-6

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