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Passive Dendrites Enable Single Neurons to Compute Linearly Non-separable Functions

Romain Daniel Cazé 1, 2, * Mark Humphries 1, 3 Boris Gutkin 1
* Corresponding author
1 Group for Neural Theory [Paris]
LNC - Laboratoire de Neurosciences cognitives, IEC - Institut d'étude de la cognition
Abstract : Local supra-linear summation of excitatory inputs occurring in pyramidal cell dendrites, the so-called dendritic spikes, results in independent spiking dendritic sub-units, which turn pyramidal neurons into two-layer neural networks capable of computing linearly non-separable functions, such as the exclusive OR. Other neuron classes, such as interneurons, may possess only a few independent dendritic sub-units, or only passive dendrites where input summation is purely sub-linear, and where dendritic sub-units are only saturating. To determine if such neurons can also compute linearly non-separable functions, we enumerate, for a given parameter range, the Boolean functions implementable by a binary neuron model with a linear sub-unit and either a single spiking or a saturating dendritic sub-unit. We then analytically generalize these numerical results to an arbitrary number of non-linear sub-units. First, we show that a single non-linear dendritic sub-unit, in addition to the somatic non-linearity, is sufficient to compute linearly non-separable functions. Second, we analytically prove that, with a sufficient number of saturating dendritic sub-units, a neuron can compute all functions computable with purely excitatory inputs. Third, we show that these linearly non-separable functions can be implemented with at least two strategies: one where a dendritic sub-unit is sufficient to trigger a somatic spike; another where somatic spiking requires the cooperation of multiple dendritic sub-units. We formally prove that implementing the latter architecture is possible with both types of dendritic sub-units whereas the former is only possible with spiking dendrites. Finally, we show how linearly non-separable functions can be computed by a generic two-compartment biophysical model and a realistic neuron model of the cerebellar stellate cell interneuron. Taken together our results demonstrate that passive dendrites are sufficient to enable neurons to compute linearly non-separable functions.
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Romain Daniel Cazé, Mark Humphries, Boris Gutkin. Passive Dendrites Enable Single Neurons to Compute Linearly Non-separable Functions. PLoS Computational Biology, Public Library of Science, 2013, 9 (2), pp.e1002867. ⟨10.1371/journal.pcbi.1002867⟩. ⟨inserm-02140108⟩



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