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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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Cerebellar stellate cell interneurons can implement the dual feature binding problem.(A) A schematic representation of the biophysical model, the circle represents the soma and the 3 cylinders correspond to dendrites, their size is expressed in m, the blue bars represent the region where the four cell assemblies , each made of 100 pre-synaptic neurons, makes contacts every m, between 30 and 100 m away from the soma.  and  make contact on the the left dendrite, whereas  and  make contact on the right dendrite. (B) Above, a spike density plot of the cell assembly , each made of 100 pre-synaptic neurons, when  (blue) or  (black). Below, the corresponding raster plot. (C) Above, the probability of a post-synaptic spike averaged over 10 trials, when two scattered inputs are active (Scat: , , , or ) or when two clustered inputs are active (Clust:  or ). The bars correspond to the variance of the binomial distribution for p(post spike). Below, somatic voltage traces in clustered (black) or in scattered (blue) condition. Note that our model of cerebellar stellate cells fires significantly more often (Binomial test, ) when inputs are scattered over the dendritic tree.
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pcbi-1002867-g005: Cerebellar stellate cell interneurons can implement the dual feature binding problem.(A) A schematic representation of the biophysical model, the circle represents the soma and the 3 cylinders correspond to dendrites, their size is expressed in m, the blue bars represent the region where the four cell assemblies , each made of 100 pre-synaptic neurons, makes contacts every m, between 30 and 100 m away from the soma. and make contact on the the left dendrite, whereas and make contact on the right dendrite. (B) Above, a spike density plot of the cell assembly , each made of 100 pre-synaptic neurons, when (blue) or (black). Below, the corresponding raster plot. (C) Above, the probability of a post-synaptic spike averaged over 10 trials, when two scattered inputs are active (Scat: , , , or ) or when two clustered inputs are active (Clust: or ). The bars correspond to the variance of the binomial distribution for p(post spike). Below, somatic voltage traces in clustered (black) or in scattered (blue) condition. Note that our model of cerebellar stellate cells fires significantly more often (Binomial test, ) when inputs are scattered over the dendritic tree.

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.


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

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

Cerebellar stellate cell interneurons can implement the dual feature binding problem.(A) A schematic representation of the biophysical model, the circle represents the soma and the 3 cylinders correspond to dendrites, their size is expressed in m, the blue bars represent the region where the four cell assemblies , each made of 100 pre-synaptic neurons, makes contacts every m, between 30 and 100 m away from the soma.  and  make contact on the the left dendrite, whereas  and  make contact on the right dendrite. (B) Above, a spike density plot of the cell assembly , each made of 100 pre-synaptic neurons, when  (blue) or  (black). Below, the corresponding raster plot. (C) Above, the probability of a post-synaptic spike averaged over 10 trials, when two scattered inputs are active (Scat: , , , or ) or when two clustered inputs are active (Clust:  or ). The bars correspond to the variance of the binomial distribution for p(post spike). Below, somatic voltage traces in clustered (black) or in scattered (blue) condition. Note that our model of cerebellar stellate cells fires significantly more often (Binomial test, ) when inputs are scattered over the dendritic tree.
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Related In: Results  -  Collection

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

pcbi-1002867-g005: Cerebellar stellate cell interneurons can implement the dual feature binding problem.(A) A schematic representation of the biophysical model, the circle represents the soma and the 3 cylinders correspond to dendrites, their size is expressed in m, the blue bars represent the region where the four cell assemblies , each made of 100 pre-synaptic neurons, makes contacts every m, between 30 and 100 m away from the soma. and make contact on the the left dendrite, whereas and make contact on the right dendrite. (B) Above, a spike density plot of the cell assembly , each made of 100 pre-synaptic neurons, when (blue) or (black). Below, the corresponding raster plot. (C) Above, the probability of a post-synaptic spike averaged over 10 trials, when two scattered inputs are active (Scat: , , , or ) or when two clustered inputs are active (Clust: or ). The bars correspond to the variance of the binomial distribution for p(post spike). Below, somatic voltage traces in clustered (black) or in scattered (blue) condition. Note that our model of cerebellar stellate cells fires significantly more often (Binomial test, ) when inputs are scattered over the dendritic tree.
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.

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