On the properties of input-to-output transformations in neuronal networks

Andrey  Olypher; Jean  Vaillant; Andrey  Olypher; Jean  Vaillant

doi:10.3934/mbe.2016009

Mathematical Biosciences and Engineering

2016, Volume 13, Issue 3: 579-596. doi: 10.3934/mbe.2016009

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On the properties of input-to-output transformations in neuronal networks

Andrey Olypher ¹,
Jean Vaillant ²

1.
School of Science and Technology, Georgia Gwinnett College, Lawrenceville, GA 30043
2.
Department of Mathematics and Informatics, Université des Antilles, Pointe-à-Pitre, Guadeloupe

Received: 01 March 2015 Accepted: 29 June 2018 Published: 01 January 2016
MSC : Primary: 92B20, 68T10; Secondary: 68M10, 05D40.

Information processing in neuronal networks in certain important cases can be considered as maps of binary vectors, where ones (spikes) and zeros (no spikes) of input neurons are transformed into spikes and no spikes of output neurons. A simple but fundamental characteristic of such a map is how it transforms distances between input vectors into distances between output vectors. We advanced earlier known results by finding an exact solution to this problem for McCulloch-Pitts neurons. The obtained explicit formulas allow for detailed analysis of how the network connectivity and neuronal excitability affect the transformation of distances in neurons. As an application, we explored a simple model of information processing in the hippocampus, a brain area critically implicated in learning and memory. We found network connectivity and neuronal excitability parameter values that optimize discrimination between similar and distinct inputs. A decrease of neuronal excitability, which in biological neurons may be associated with decreased inhibition, impaired the optimality of discrimination.
- pattern discrimination,
- McCulloch-Pitts neurons.,
- neuronal networks,
- Information processing,
- hippocampus
Citation: Andrey Olypher, Jean Vaillant. On the properties of input-to-output transformations in neuronal networks[J]. Mathematical Biosciences and Engineering, 2016, 13(3): 579-596. doi: 10.3934/mbe.2016009

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Abstract

Information processing in neuronal networks in certain important cases can be considered as maps of binary vectors, where ones (spikes) and zeros (no spikes) of input neurons are transformed into spikes and no spikes of output neurons. A simple but fundamental characteristic of such a map is how it transforms distances between input vectors into distances between output vectors. We advanced earlier known results by finding an exact solution to this problem for McCulloch-Pitts neurons. The obtained explicit formulas allow for detailed analysis of how the network connectivity and neuronal excitability affect the transformation of distances in neurons. As an application, we explored a simple model of information processing in the hippocampus, a brain area critically implicated in learning and memory. We found network connectivity and neuronal excitability parameter values that optimize discrimination between similar and distinct inputs. A decrease of neuronal excitability, which in biological neurons may be associated with decreased inhibition, impaired the optimality of discrimination.

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