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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 types of local dendritic non-linearities.(A–B) The x-axis (Expected EPSP) is the arithmetic sum of two EPSPs induced by two distinct stimulations and y-axis (Measured EPSP) is the measured EPSP when the stimulations are made simultaneously. (A) Observations made on pyramidal neurons (redrawn from [13]). Summation is supra-linear and sub-linear due to the occurrence of a dendritic spike. (B) Â Observations made on cerebellar interneurons (redrawn from [10]). In this case summation is purely sub-linear due to a saturation caused by a reduced driving force. (C) Â The activation function modeling the dendritic spike type non-linear summation: both supra-linear and sub-linear on . (D) Â The activation function modeling the saturation type non-linear summation: strictly sub-linear on . (E) Structure and parameters of the neuron model:  and  are binary variables describing pre and post-synaptic neuronal activity; in circles are two independent sets of non-negative integer-valued synaptic weights respectively for the linear (black) and the non-linear integration (blue) sub-units; in the blue square,  and  are the non-negative integer-valued threshold and height that parameterize the dendritic activation function ; in the black square  is a positive integer-valued threshold determining post-synaptic firing. (F) Â Truth tables of three Boolean functions for  inputs: AND, NAND, and XOR. The first column gives the possible values of the input vector ; the other three columns give the binary outputs  in response to each  for the three functions considered.
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pcbi-1002867-g001: Two types of local dendritic non-linearities.(A–B) The x-axis (Expected EPSP) is the arithmetic sum of two EPSPs induced by two distinct stimulations and y-axis (Measured EPSP) is the measured EPSP when the stimulations are made simultaneously. (A) Observations made on pyramidal neurons (redrawn from [13]). Summation is supra-linear and sub-linear due to the occurrence of a dendritic spike. (B) Â Observations made on cerebellar interneurons (redrawn from [10]). In this case summation is purely sub-linear due to a saturation caused by a reduced driving force. (C) Â The activation function modeling the dendritic spike type non-linear summation: both supra-linear and sub-linear on . (D) Â The activation function modeling the saturation type non-linear summation: strictly sub-linear on . (E) Structure and parameters of the neuron model: and are binary variables describing pre and post-synaptic neuronal activity; in circles are two independent sets of non-negative integer-valued synaptic weights respectively for the linear (black) and the non-linear integration (blue) sub-units; in the blue square, and are the non-negative integer-valued threshold and height that parameterize the dendritic activation function ; in the black square is a positive integer-valued threshold determining post-synaptic firing. (F) Â Truth tables of three Boolean functions for inputs: AND, NAND, and XOR. The first column gives the possible values of the input vector ; the other three columns give the binary outputs in response to each for the three functions considered.

Mentions: Supra-linear summation, the dendritic spikes, has been described for a variety of active dendritic mechanisms. For this type of local summation the measured EPSP peak is first above then below the expected arithmetic sum of EPSPs as shown on Figure 1A. Synapse-driven membrane potential depolarization can open [3], [4], [5], [6], or NMDA receptor [6], [7], [8], [9] channels sufficiently to amplify the initial depolarization, and evoke a dendritic spike.


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

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

Two types of local dendritic non-linearities.(A–B) The x-axis (Expected EPSP) is the arithmetic sum of two EPSPs induced by two distinct stimulations and y-axis (Measured EPSP) is the measured EPSP when the stimulations are made simultaneously. (A) Observations made on pyramidal neurons (redrawn from [13]). Summation is supra-linear and sub-linear due to the occurrence of a dendritic spike. (B) Â Observations made on cerebellar interneurons (redrawn from [10]). In this case summation is purely sub-linear due to a saturation caused by a reduced driving force. (C) Â The activation function modeling the dendritic spike type non-linear summation: both supra-linear and sub-linear on . (D) Â The activation function modeling the saturation type non-linear summation: strictly sub-linear on . (E) Structure and parameters of the neuron model:  and  are binary variables describing pre and post-synaptic neuronal activity; in circles are two independent sets of non-negative integer-valued synaptic weights respectively for the linear (black) and the non-linear integration (blue) sub-units; in the blue square,  and  are the non-negative integer-valued threshold and height that parameterize the dendritic activation function ; in the black square  is a positive integer-valued threshold determining post-synaptic firing. (F) Â Truth tables of three Boolean functions for  inputs: AND, NAND, and XOR. The first column gives the possible values of the input vector ; the other three columns give the binary outputs  in response to each  for the three functions considered.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1002867-g001: Two types of local dendritic non-linearities.(A–B) The x-axis (Expected EPSP) is the arithmetic sum of two EPSPs induced by two distinct stimulations and y-axis (Measured EPSP) is the measured EPSP when the stimulations are made simultaneously. (A) Observations made on pyramidal neurons (redrawn from [13]). Summation is supra-linear and sub-linear due to the occurrence of a dendritic spike. (B) Â Observations made on cerebellar interneurons (redrawn from [10]). In this case summation is purely sub-linear due to a saturation caused by a reduced driving force. (C) Â The activation function modeling the dendritic spike type non-linear summation: both supra-linear and sub-linear on . (D) Â The activation function modeling the saturation type non-linear summation: strictly sub-linear on . (E) Structure and parameters of the neuron model: and are binary variables describing pre and post-synaptic neuronal activity; in circles are two independent sets of non-negative integer-valued synaptic weights respectively for the linear (black) and the non-linear integration (blue) sub-units; in the blue square, and are the non-negative integer-valued threshold and height that parameterize the dendritic activation function ; in the black square is a positive integer-valued threshold determining post-synaptic firing. (F) Â Truth tables of three Boolean functions for inputs: AND, NAND, and XOR. The first column gives the possible values of the input vector ; the other three columns give the binary outputs in response to each for the three functions considered.
Mentions: Supra-linear summation, the dendritic spikes, has been described for a variety of active dendritic mechanisms. For this type of local summation the measured EPSP peak is first above then below the expected arithmetic sum of EPSPs as shown on Figure 1A. Synapse-driven membrane potential depolarization can open [3], [4], [5], [6], or NMDA receptor [6], [7], [8], [9] channels sufficiently to amplify the initial depolarization, and evoke a dendritic spike.

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