## Optimization of connection patterns in networks of oscillators

##### Resumen

The means by which pacemaker cells of the mammalian suprachiasmatic nucleus (SCN) are synchronized is unknown. In the absence of anatomical data on the interneuronal connections among SCN neurons, we have modeled the SCN network in terms of a number of possible connection topologies. We employ a mathematical model proposed by Achermann and Kunz (1999), to study the problem of interpreting synchronization in the SCN network from a dynamical systems viewpoint. We vary the proportion of local or nearest neighbor neuronal connections and global or long distance connections in the SCN, and compare time elapsed before synchronization is established. Time of resynchronization is the time elapsed before SCN neurons reestablish their phase-locked circadian response after complete randomization of initial phases. We consider two models where one, is a three-dimensional model with 8000 neurons connected as a torus and the second, an one-dimensional model consisting of 500 neurons (mean period=24 hr, S.D.=1 hr), each with Kronauer dynamics, with weak inhibitory coupling to each other, similar to the model described by Achermann and Kunz. Neurons are arranged in a ring in all directions, and connections are assumed to be symmetric with respect to cell locations.
Mainly we studied two different dynamics: a three dimensional model of 8000 neurons with fixed connection patterns and one-dimensional model of 500 neurons with random symmetric connection patterns. The three dimensional models was simulated under four different light conditions: Absence of light, presence of light,10x10x10 core light and 16x16x16 core light. It was observed that the existence of few long distance connections make significant difference in the synchronization times. For instance, the synchronization times for pure locally connected and few long distance connected networks under the core light of 16x16x16 were 17 and 11 days, respectively. Results were similar for the other light conditions. In the second stage, we studied the synchronization and phase locking times of two types of randomly (symmetric) connected networks: Purely local and few long distance connections networks. It was clearly observed, with a light increase in long distance connections the phase locking time drops dramatically. For instance, if the number of long distance connections are increased from 2 to 10 (out of 30 connections), the phase locking time drops from 47 days to 6 days. In conjunction with our previous finding that completely interconnected global networks resynchronize much more quickly than the physiological oscillator in the SCN, these results suggest the possibility that the SCN topology is a "small world" network, i.e., a neuronal network with largely local interconnections plus a small number of long distance connections.