Abstracts

Rationale: The brain is a highly recurrent network with enumerable positive and negative feedback loops. Only the smallest most carefully tuned first-order linear dynamical systems tend be stable. Therefore, it is remarkable that most brains maintain a homeostatic level of activity and avoid seizure. Either connectivity is exquisitely regulated or there are important stabilizing properties within neural networks. Adaptation results in decreased neuronal response over time to sustained input. Potential mechanisms include spike rate adaptation and synaptic depression. We previously found that excitatory neurons in the thalamus and cortex exhibit multiple timescale spike rate adaptation that can be well-modeled using fractional order calculus. (Lundstrom 2008, 2010) Here, our goal is to explore the stability of neural networks with adaptation implemented using fractional dynamics.

Methods: We directly compared neural networks with and without adaptation. The connectivity between excitatory and inhibitory nodes was assigned from a gaussian random distribution to explore the extreme case where connectivity is untuned. We mathematically derived a finite time approximation of fractional integral systems that enabled direct eigenanalysis of stability, i.e., positive eigenvalues indicate unstable systems while negative eigenvalues indicate stability.

Results: Positive self-connections, which create positive feedback and exponential runaway in first-order systems, were instead a stabilizing motif for fractional systems (FIgures 1B, 2B-C). Large first-order neural networks without adaptation are clearly unstable. In comparison, large fractional systems had both fewer and smaller positive eigenvalues, usually near zero (Figure 1B), suggesting that multiple timescale adaptation keeps large networks near the edge of instability, and therefore slightly chaotic. While the excitation:inhibition (E:I) ratio strongly determines the stability of first-order systems, fractional systems were far less sensitive the E:I ratio.

Figure 1. Networks with positive feedback (A) Are usually unstable. (B) Our eigenanalysis shows fractional integral systems have improved stability.

Figure 2. The network (Figure 1A) Diverges without adaptation (B), but remains stable (C) with adaption when driven with an external stimulus (A).

Conclusions: Multiple timescale adaptation improves the stability of even randomly connected networks with positive feedback and reduces, but does not eliminate, sensitivity to inhibition levels. Previous intuitions about positive feedback loops and E:I ratios may require reconsideration for systems with fractional dynamics. Since at least some brain networks likely exhibit fractional dynamics, these results may eventually help diagnose and treat epilepsy.

References:_x000D_ Lundstrom BN, Higgs MH, Spain WJ, Fairhall AL. Fractional differentiation by neocortical pyramidal neurons. Nat. Neurosci. 2008;11:1335.
Lundstrom BN, Fairhall AL, Maravall M. Multiple timescale encoding of slowly varying whisker stimulus envelope in cortical and thalamic neurons in vivo. J. Neurosci. 2010;30:5071-5077.

Funding: BNL was funded by NIH NINDS K23NS112339.