Passive dendrites enable single neurons to compute linearly non-separable functions.
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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.We then analytically generalize these numerical results to an arbitrary number of non-linear sub-units.Taken together our results demonstrate that passive dendrites are sufficient to enable neurons to compute linearly non-separable functions.
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Affiliation: Group for Neural Theory, INSERM U960, Ecole Normale Superieure, Paris, France. romain.caze@ens.fr
ABSTRACT
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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. Related in: MedlinePlus |
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Mentions: Next, we looked at how a realistic biophysical model can implement a linearly non-separable positive Boolean function. Figure 5 shows how the dFBP can be implemented in a realistic model of cerebellar stellate cell. This cell type is interesting because it is an experimentally-studied example of an electrotonically compact neuron with ‘passive’ dendrites [10]. Moreover, the same study demonstrated using patch-clamp and simultaneous glutamate uncaging that integration is strictly sub-linear in the distal dendritic region (30 m away from the soma). These features make the cerebellar stellate cell an ideal candidate to see if our binary model leads to testable experimental predictions. |
View Article: PubMed Central - PubMed
Affiliation: Group for Neural Theory, INSERM U960, Ecole Normale Superieure, Paris, France. romain.caze@ens.fr