Sensitivity and stability : A signal propagation sweet spot in a sheet

In this paper we demonstr imisedwhen two conditio Second, the network’s top nodes, yet lies within the which to describe this reg these results to examine t networks. 1. Signal Propagation Signal propagation is central to the control systems of all adaptive agents in that it is crucial for the effective transduction of sensory input intomotoroutput. Biological systemsseemtoachieve Please cite this article in press as: Buckley, C.L., Bullock, S., Sensitivity and s crossing neurons. BioSystems (2008), doi:10.1016/j.biosystems.2008.05.02 successful signal propagation over extended networks of neurons with relative ease. Feedforward neural architectures have been employed to investigate how signals propagate across networks and can construct complex mappings between input and output (Litvak et al., 2003).However, in general, biological neural networks are recurrent, even in systems that have previously been idealised as feedforward in nature, e.g., the columns within the visual cortex have recurrent connectionswithin andbetween layers (Carlson, 1991). Signal propagation across such recurrent networks is likely to bemore complex than in feedforwardnetworks,where it is taken for granted. There has been a deal of speculation in neuroscience concerning mechanisms that could promote signal propagation across a sequence of neurons (Turrigiano, 1999). One set of ideas involves the behaviour of nodes that tend to interact at the centre of their operating ranges. In general, networks of such neurons are thought to be computationally rich. More specifically, in this regime, nodes are maximally sensitive to input, potentially facilitating more efficient signal propagation across extended networks. Moreover, Turrigiano (1999) describes how homeostatic processes (HPs) might ∗ Corresponding author. E-mail address: [email protected] (C.L. Buckley). 0303-2647/$ – see front matter © 2008 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2008.05.026 BJ, UK at signal propagation across a laminar sheet of recurrent neurons is maxmet. First, neuronsmust be in the so-called centre crossing configuration. and weights must be such that the network comprises strongly coupled kly coupled regime. We develop tools from linear stability analysis with in terms of the connectivity and weight strengths of a network. We use parent tension between the sensitivity and instability of centre crossing © 2008 Elsevier Ireland Ltd. All rights reserved. actively “keep neurons at the centre of their operating ranges” (Turrigiano, 1999). Inspired by this work, Williams (2006) studied how an abstraction of these HPs affected the ability of a continuous time recurrent neural network (CTRNN) to propagate signals. In this work, HP provided a simple feedback mechanism that altered the gain and bias of a node such that its input tended to lie at the centre of its transfer function. He hypothesized that networks composed of such nodes tability: A signal propagation sweet spot in a sheet of recurrent centre 6 would be better able propagate signals, because localHP at the level of individual nodes would drive networks into the most sensitive region of their dynamics. Williams found that HP drove systems toward a configuration that has been identified as significant within the CTRNN literature. In this so-called “centre crossing configuration” all nodes in a CTRNN interact at the centre of their sigmoid transfer functions (Mathayomchan and Beer, 2002), a mathematical property that bears close resemblance to the biological ideas highlighted by Turrigiano, amongst others. Williams also demonstrated that signal propagation was improved within such centre crossing networks. However, this signal propagation was impoverished within larger networks, and did not approach the performance achieved by an equivalent feedforward architecture even for small networks (personal communication). One possible reason for these results can be induced from the original work on centre crossing CTRNNs (Mathayomchan and Beer, 2002). Here, it was demonstrated that the generation of rhythmic patterns evolved more readily in such networks. This is due to the fact that centre crossing networks are likely to produce oscillatory dynamics. Such oscillatory behaviour is likely to corrupt the transmission of signals across extended net-