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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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A dendritic non-linearity enables the computation of linearly non-separable Boolean functions.(A) Number of computable representative positive Boolean functions depending on the number of input variables  and on the type of synaptic integration: purely linear (lin∶black), linear with a spiking dendritic sub-unit (spk∶green), linear with saturating dendritic sub-unit (sat∶blue). In red is the maximal number of positive representative functions computable for a given , this number is taken from [37] as the number of functions in condition lin (black). Upper panel: number of computable functions (in bold are lower bounds); lower panel: summary bar charts on logarithmic scale. (B) Venn diagram for the sets of Boolean functions for . The set border color depends on the type of integration, as per panel A (relative size of sets not to scale). Stars are examples of Boolean functions within each set.
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pcbi-1002867-g002: A dendritic non-linearity enables the computation of linearly non-separable Boolean functions.(A) Number of computable representative positive Boolean functions depending on the number of input variables and on the type of synaptic integration: purely linear (lin∶black), linear with a spiking dendritic sub-unit (spk∶green), linear with saturating dendritic sub-unit (sat∶blue). In red is the maximal number of positive representative functions computable for a given , this number is taken from [37] as the number of functions in condition lin (black). Upper panel: number of computable functions (in bold are lower bounds); lower panel: summary bar charts on logarithmic scale. (B) Venn diagram for the sets of Boolean functions for . The set border color depends on the type of integration, as per panel A (relative size of sets not to scale). Stars are examples of Boolean functions within each set.

Mentions: To determine the computational capacity, we counted only the computable representative Boolean functions (see Materials and Methods Boolean Algebra for formal definition). Moreover, we controlled the parameter searches using two analytically known sizes of Boolean function sets: first, the size of the set of all representative positive Boolean functions [34], [37], known for a number of binary variables up to 6; second, within this set of functions, the number of linearly separable representative Boolean functions [37]. This last number corresponded to the exact computational capacity for the purely linear model (). Therefore by comparing to these two known sizes we could see if the model including a dendritic non-linearity ( or ) enabled computation of linearly non-separable functions and, if so, what proportion of those functions could be accessed. The relationship between these sets of functions and the set that can be accessed by a model including a non-linear dendritic sub-unit is illustrated schematically in Figure 2B.


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

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

A dendritic non-linearity enables the computation of linearly non-separable Boolean functions.(A) Number of computable representative positive Boolean functions depending on the number of input variables  and on the type of synaptic integration: purely linear (lin∶black), linear with a spiking dendritic sub-unit (spk∶green), linear with saturating dendritic sub-unit (sat∶blue). In red is the maximal number of positive representative functions computable for a given , this number is taken from [37] as the number of functions in condition lin (black). Upper panel: number of computable functions (in bold are lower bounds); lower panel: summary bar charts on logarithmic scale. (B) Venn diagram for the sets of Boolean functions for . The set border color depends on the type of integration, as per panel A (relative size of sets not to scale). Stars are examples of Boolean functions within each set.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1002867-g002: A dendritic non-linearity enables the computation of linearly non-separable Boolean functions.(A) Number of computable representative positive Boolean functions depending on the number of input variables and on the type of synaptic integration: purely linear (lin∶black), linear with a spiking dendritic sub-unit (spk∶green), linear with saturating dendritic sub-unit (sat∶blue). In red is the maximal number of positive representative functions computable for a given , this number is taken from [37] as the number of functions in condition lin (black). Upper panel: number of computable functions (in bold are lower bounds); lower panel: summary bar charts on logarithmic scale. (B) Venn diagram for the sets of Boolean functions for . The set border color depends on the type of integration, as per panel A (relative size of sets not to scale). Stars are examples of Boolean functions within each set.
Mentions: To determine the computational capacity, we counted only the computable representative Boolean functions (see Materials and Methods Boolean Algebra for formal definition). Moreover, we controlled the parameter searches using two analytically known sizes of Boolean function sets: first, the size of the set of all representative positive Boolean functions [34], [37], known for a number of binary variables up to 6; second, within this set of functions, the number of linearly separable representative Boolean functions [37]. This last number corresponded to the exact computational capacity for the purely linear model (). Therefore by comparing to these two known sizes we could see if the model including a dendritic non-linearity ( or ) enabled computation of linearly non-separable functions and, if so, what proportion of those functions could be accessed. The relationship between these sets of functions and the set that can be accessed by a model including a non-linear dendritic sub-unit is illustrated schematically in Figure 2B.

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