Introduction
Neuronal networks (NNs) are predominantly modeled as dynamical systems, requiring ad hoc analyses to explore the interplay between population codes (PCs) and computation [3,4]. By viewing PCs as distributions over the NN state space [2] we develop a framework for modeling emergent computation through the lens of multi-agent (MA) optimal control (OC). In it, neurons regulate their local parameters to collectively control the PC and (in turn) optimize a cost function. A mean-field (MF) limit is derived for a large class of NNs, whence we prove theorems establishing global optima as laws of self-organization. Such models are built on a rich mathematical literature, with great potential for further theoretical results and learning algorithms.
Methods
Mean-field game/control theory is an analytical tool for characterizing/learning optimal decision-making in complex strategic systems. MF limits are useful because they ‘average out’ microscale fluctutations to distill macroscale behavior, dramatically simplifying controls while providing powerful approximations for large, finite populations. However, traditional MF theories fail for general networks. We resolve this limitation and maximize the class of compatible models/cost functions while preserving a detailed theoretical characterization. The setup in [1] closely resembles what we propose, but lacks control. Moreover, the MF is taken to model a PC by characterizing the NN up to each neuron’s identity.
Results
For a large class of computational tasks where neurons contribute to the PC anonymously, we characterize OCs without any a priori restriction on their structure or the information available to each neuron. Specifically, under an OC: (i) neurons are decentralized (acting independently given their local state and each subpopulation’s MF) such that the emergent parameter regime is realized through a process of self-organization, and (ii) neurons of the same species deploy an identical strategy, reinforcing its biological plausibility; neurons of the same type will behave identically under fixed conditions. As a concrete example, we construct a multi-population Hodgkin Huxley network designed to express a binary decision via its MF PC.
Discussion
Reformulating NNs as MF MA systems unlocks a wealth of analytical tools, learning algorithms and theoretical guarantees ripe for neuroscience applications. This work is a crucial first step in bridging the gap. Our technical results are complemented by the conjecture that PCs are expressed through the MF, which in turn emerges from optimal laws of self-organization at the microscale. Unlike many cost function-based NNs, these are global optima, affording greater normative potential. In exchange, models require target computations to be expressed as explicit cost functions assessing the PC directly, which has received little attention to-date. Future work will focus on developing this aspect further, along with simulation/learning studies.
References
1. Baladron, J., Fasoli, D., Faugeras, O., & Touboul, J. (2012). Mean-field description and propagation of chaos in networks of hodgkin-huxley and fitzhugh-nagumo neurons. Journal of Mathematical Neuroscience, 2, 10. doi: 10.1186/2190-8567-2-10
2. Beck, J. M., Latham, P. E., & Pouget, A. (2011). Marginalization in neural circuits with divisive normalization. The Journal of Neuroscience, 31(43), 15310–15319. doi: 10.1523/JNEUROSCI.1706-11.2011
3. Denève, S., & Machens, C. K. (2016). Efficient codes and balanced networks. Nature Neuroscience, 19(3), 375–382. doi: 10.1038/nn.4243
4. Wong, K.-F., & Wang, X.-J. (2006). A recurrent network mechanism of time integration in perceptual decisions. The Journal of Neuroscience, 26(4), 1314–1328. doi: 10.1523/JNEUROSCI.3733-05.2006
Acknowledgement
We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC).
\nNous remercions le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG) de son soutien.
