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Cognitive Neurodynamics
Published in: Cognitive Neurodynamics 6/2012

01-12-2012 | Research Article

Combined effects of LTP/LTD and synaptic scaling in formation of discrete and line attractors with persistent activity from non-trivial baseline

Authors: Timothee Leleu, Kazuyuki Aihara

Published in: Cognitive Neurodynamics | Issue 6/2012

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Abstract

In this article, we analyze combined effects of LTP/LTD and synaptic scaling and study the creation of persistent activity from a periodic or chaotic baseline attractor. The bifurcations leading to the creation of new attractors have been detailed; this was achieved using a mean field approximation. Attractors encoding persistent activity can notably appear via generalized period-doubling bifurcations, tangent bifurcations of the second iterates or boundary crises, after which the basins of attraction become irregular. Synaptic scaling is shown to maintain the coexistence of a state of persistent activity and the baseline. According to the rate of change of the external inputs, different types of attractors can be formed: line attractors for rapidly changing external inputs and discrete attractors for constant external inputs.
previous article Identifying type I excitability using dynamics of stochastic neural firing patterns
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Appendix
Available only for authorised users
Footnotes
1
It is important to note that phenomena that are related to spike timing (such as synchrony of spikes) are not taken into account here; however, they may play an important role in cognition studies (a temporal-coding hypothesis). Synchronous firing may disrupt persistent activity (Compte 2006) because of the different time scales for the receptors (primarily AMPA and GABAA), and create oscillations (Compte et al. 2003). All receptors are assumed to have an identical time scale for simplicity; thus only asynchronous persistent activity is considered in the following.
 
2
Note that the chaos observed in this network is of a different nature than the one observed in Sompolinsky et al. (1988), Doyon et al. (1994). Here, the average activity over the population of neurons, which is given by the mean field approximation, is chaotic (see Appendix 4).
 
3
Otherwise, only LTP/LTD are active, see Eq. (7).
 
4
An infinite set of maps indexed by the step n are thus considered.
 
5
A soft bifurcation induces new stable attractors in a small neighborhood of the old one. For example, tangent bifurcations far from the cusp and subcritical pitchfork bifurcations are hard, whereas supercritical pitchfork bifurcations are soft (Hoppensteadt and Izhikevich 1997).
 
6
For a normally hyperbolic invariant manifold (NHIM), the contraction vectors orthogonal to the manifold are stronger than those along the manifold (Hoppensteadt and Izhikevich 1997). NHIM can be interpreted as the generalization of a hyperbolic fixed point to non-trivial attractors. Formal definitions can be found in Fenichel (1972); Hirsch and Shub (1977), Pesin (2004). A fundamental property of these manifolds is that they are persistent under perturbations, i.e., the perturbed invariant manifold has normal and tangent subspaces that are close to the original manifold. This structural stability assures that an attractor of the perturbed system lies near an attractor of the unperturbed system.
 
7
Note that the assumption in Eq. (19) does not hold for \(\Updelta^{T_s} X_i(n), T_s \ll T_0\). Indeed, the terms X i (n) and X i (n − T s ) average almost identical trajectories, and therefore do not describe the attractors of two different dynamical systems (perturbed \(\tilde{G}(n)\) and unperturbed \(\tilde{G}(n-T_1)\)), but the difference between consecutive steps of the same attractor. In that case, these consecutive steps could be at a distance that is equal to the size of the attractor.
 
8
These populations should not be confused with excitatory and inhibitory populations which are common in neuroscience; for example in the Wilson-Conwan model (1972).
 
9
For simplicity, we assume γ+ =  − γ− = γ > 0. Similar dynamics to the ones described in the following are also observed for other choices of γ− and are not restrained to this set of parameters.
 
10
These two curves are the equivalent of nullclines for a system of differential equations.
 
11
Other distribution functions that induce multiple discrete attractors could be considered. This article focus only on the uniform distribution to show the creation of line attractors.
 
12
Simply considering the symmetry of the change in weights, all neurons are similarly affected by synaptic plasticity.
 
13
“Complex” refers here to the Kolmogorov-Sinai entropy.
 
14
The creation of fractal basin boundaries is also observed for systems with continuous time (Ott 1993), and is not an artifact due to the discrete time used in the current article.
 
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Metadata
Title
Combined effects of LTP/LTD and synaptic scaling in formation of discrete and line attractors with persistent activity from non-trivial baseline
Authors
Timothee Leleu
Kazuyuki Aihara
Publication date
01-12-2012
Publisher
Springer Netherlands
Published in
Cognitive Neurodynamics / Issue 6/2012
Print ISSN: 1871-4080
Electronic ISSN: 1871-4099
DOI
https://doi.org/10.1007/s11571-012-9211-3

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