Abstract
In this chapter we describe a continuum model for the cortex that includes both axon-to-dendrite chemical synapses and direct neuron-to-neuron gap-junction diffusive synapses. The effectiveness of chemical synapses is determined by the voltage of the receiving dendrite V relative to its Nernst reversal potential \(V^{\rm rev}{}\). Here we explore two alternative strategies for incorporating dendritic reversal potentials, and uncover surprising differences in their stability properties and model dynamics. In the “slow-soma” variant, the \((V^{\rm rev} - V)\) weighting is applied after the input flux has been integrated at the dendrite, while for “fast-soma”, the weighting is applied directly to the input flux, prior to dendritic integration. For the slow-soma case, we find that–-provided the inhibitory diffusion (via gap-junctions) is sufficiently strong–-the cortex generates stationary Turing patterns of cortical activity. In contrast, the fast-soma destabilizes in favor of standing-wave spatial structures that oscillate at low-gamma frequency (\(\sim\)30-Hz); these spatial patterns broaden and weaken as diffusive coupling increases, and disappear altogether at moderate levels of diffusion. We speculate that the slow- and fast-soma models might correspond respectively to the idling and active modes of the cortex, with slow-soma patterns providing the default background state, and emergence of gamma oscillations in the fast-soma case signaling the transition into the cognitive state.
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Notes
- 1.
- 2.
Later we simplify the subscripting notation for diffusion so that (D ee , D ii) \(\equiv\) (D 1, D 2).
- 3.
The 2-D circular convolution algorithm was written by David Young, Department of Informatics, University of Sussex, UK. His convolve2() matlab function can be downloaded from The MathWorks File Exchange, www.mathworks.com/matlabcentral/fileexchange.
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Acknowledgment
We thank Chris Rennie for helpful discussions on convolution formulations for the cortex. This research was supported by the Royal Society of New Zealand Marsden Fund, contract 07-UOW-037.
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Steyn-Ross, M., Steyn-Ross, D., Wilson, M., Sleigh, J. (2010). Cortical patterns and gamma genesis are modulated by reversal potentials and gap-junction diffusion. In: Steyn-Ross, D., Steyn-Ross, M. (eds) Modeling Phase Transitions in the Brain. Springer Series in Computational Neuroscience, vol 4. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0796-7_12
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