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Passive dendrites enable single neurons to compute linearly non-separable functions.

Cazé RD, Humphries M, Gutkin B - PLoS Comput. Biol. (2013)

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Group for Neural Theory, INSERM U960, Ecole Normale Superieure, Paris, France. romain.caze@ens.fr

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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Two strategies to implement a linearly non-separable function.On top, the name of two possible strategies to implement the feature binding problem (FBP) based either on its DNF or CNF expression: the colored part of these expressions is the term or the clauses implemented by the dendritic sub-unit. Below, three schematics which represent parameter sets implementing FBP using either a spiking (green) or a saturating (blue) dendritic sub-unit. In circles are the value of synaptic weights (Black∶linear, green∶spiking, blue∶saturating); in colored squares (green∶spiking; blue∶saturating) are the parameters of the dendritic activation function [threshold;height], in black squares is the threshold  of the somatic sub-unit. Left, the local implementation strategy; Right, the global implementation strategy; note that a neuron cannot implement the FBP using the local strategy with a saturating dendritic sub-unit. Bottom, truth tables where the  column is the input vectors,  columns describe the neuron's input-output function, here the FBP. The int. column is the result of synaptic integration of each dendritic sub-unit (black∶linear, green∶spiking, blue∶saturating). In bold and italic are the maximum possible outputs for each sub-unit, note that for the global strategy a maximal output from a dendritic sub-unit may not trigger a somatic spike.
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pcbi-1002867-g003: Two strategies to implement a linearly non-separable function.On top, the name of two possible strategies to implement the feature binding problem (FBP) based either on its DNF or CNF expression: the colored part of these expressions is the term or the clauses implemented by the dendritic sub-unit. Below, three schematics which represent parameter sets implementing FBP using either a spiking (green) or a saturating (blue) dendritic sub-unit. In circles are the value of synaptic weights (Black∶linear, green∶spiking, blue∶saturating); in colored squares (green∶spiking; blue∶saturating) are the parameters of the dendritic activation function [threshold;height], in black squares is the threshold of the somatic sub-unit. Left, the local implementation strategy; Right, the global implementation strategy; note that a neuron cannot implement the FBP using the local strategy with a saturating dendritic sub-unit. Bottom, truth tables where the column is the input vectors, columns describe the neuron's input-output function, here the FBP. The int. column is the result of synaptic integration of each dendritic sub-unit (black∶linear, green∶spiking, blue∶saturating). In bold and italic are the maximum possible outputs for each sub-unit, note that for the global strategy a maximal output from a dendritic sub-unit may not trigger a somatic spike.

Mentions: Our numerical analysis found multiple parameter sets that could implement the three functions using either a local or a global strategy. Figure 3 (left) shows an example of how a neuron with a spiking dendritic sub-unit can compute the FBP function using a local strategy. In this strategy the neuron's output is triggered either because of the direct stimulation of the somatic sub-unit (to input vector ), or because of a dendritic spike produced by the dendritic sub-unit (to input vector ); in both cases the other sub-unit contributed nothing to the whole neuron's output. Figure 3 (right) shows an example of how a neuron with a spiking sub-unit can compute the FBP function using a global strategy. In this case an input vector triggers an output spike when both the somatic and dendritic sub-units are stimulated simultaneously. Examples of a neuron with a single spiking dendritic sub-unit implementing the dFBP and pFBP using either a local or a global strategy are shown respectively in Figure S1 and Figure S2 in Text S1. Thus a neuron with a single additional spiking dendritic sub-unit can solve binding problems using either a local or a global implementation strategy.


Passive dendrites enable single neurons to compute linearly non-separable functions.

Cazé RD, Humphries M, Gutkin B - PLoS Comput. Biol. (2013)

Two strategies to implement a linearly non-separable function.On top, the name of two possible strategies to implement the feature binding problem (FBP) based either on its DNF or CNF expression: the colored part of these expressions is the term or the clauses implemented by the dendritic sub-unit. Below, three schematics which represent parameter sets implementing FBP using either a spiking (green) or a saturating (blue) dendritic sub-unit. In circles are the value of synaptic weights (Black∶linear, green∶spiking, blue∶saturating); in colored squares (green∶spiking; blue∶saturating) are the parameters of the dendritic activation function [threshold;height], in black squares is the threshold  of the somatic sub-unit. Left, the local implementation strategy; Right, the global implementation strategy; note that a neuron cannot implement the FBP using the local strategy with a saturating dendritic sub-unit. Bottom, truth tables where the  column is the input vectors,  columns describe the neuron's input-output function, here the FBP. The int. column is the result of synaptic integration of each dendritic sub-unit (black∶linear, green∶spiking, blue∶saturating). In bold and italic are the maximum possible outputs for each sub-unit, note that for the global strategy a maximal output from a dendritic sub-unit may not trigger a somatic spike.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3585427&req=5

pcbi-1002867-g003: Two strategies to implement a linearly non-separable function.On top, the name of two possible strategies to implement the feature binding problem (FBP) based either on its DNF or CNF expression: the colored part of these expressions is the term or the clauses implemented by the dendritic sub-unit. Below, three schematics which represent parameter sets implementing FBP using either a spiking (green) or a saturating (blue) dendritic sub-unit. In circles are the value of synaptic weights (Black∶linear, green∶spiking, blue∶saturating); in colored squares (green∶spiking; blue∶saturating) are the parameters of the dendritic activation function [threshold;height], in black squares is the threshold of the somatic sub-unit. Left, the local implementation strategy; Right, the global implementation strategy; note that a neuron cannot implement the FBP using the local strategy with a saturating dendritic sub-unit. Bottom, truth tables where the column is the input vectors, columns describe the neuron's input-output function, here the FBP. The int. column is the result of synaptic integration of each dendritic sub-unit (black∶linear, green∶spiking, blue∶saturating). In bold and italic are the maximum possible outputs for each sub-unit, note that for the global strategy a maximal output from a dendritic sub-unit may not trigger a somatic spike.
Mentions: Our numerical analysis found multiple parameter sets that could implement the three functions using either a local or a global strategy. Figure 3 (left) shows an example of how a neuron with a spiking dendritic sub-unit can compute the FBP function using a local strategy. In this strategy the neuron's output is triggered either because of the direct stimulation of the somatic sub-unit (to input vector ), or because of a dendritic spike produced by the dendritic sub-unit (to input vector ); in both cases the other sub-unit contributed nothing to the whole neuron's output. Figure 3 (right) shows an example of how a neuron with a spiking sub-unit can compute the FBP function using a global strategy. In this case an input vector triggers an output spike when both the somatic and dendritic sub-units are stimulated simultaneously. Examples of a neuron with a single spiking dendritic sub-unit implementing the dFBP and pFBP using either a local or a global strategy are shown respectively in Figure S1 and Figure S2 in Text S1. Thus a neuron with a single additional spiking dendritic sub-unit can solve binding problems using either a local or a global implementation strategy.

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Group for Neural Theory, INSERM U960, Ecole Normale Superieure, Paris, France. romain.caze@ens.fr

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.

Show MeSH
Related in: MedlinePlus