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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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Reduced biophysical models can implement linearly non-separable functions using dendritic saturations or dendritic spikes.(A) The biophysical model. Upper panel: schematic representation of the biophysical model, where synaptic inputs are clustered (Prox and Dist), and its morphological parameters (input locations, diameters, and dendritic length are in m). Below, expected arithmetic sum versus measured somatic EPSPs: for peri-somatic AMPA stimulation (black dots) producing a linear EPSP integration; for distal NMDA stimulation (green dots) producing a spiking type non-linear summation; for distal AMPA stimulation (in blue) producing a saturation type non-linear summation. (B) Implementation of the feature binding problem using NMDA receptor synapses for the distal dendritic region, illustrating the DNF/local strategy of synaptic placement. Top panel shows how each input makes synaptic contacts in a 10  zone either on the peri-somatic or on the distal dendritic region. Below, voltage traces (black∶soma; green∶distal dendrites) in response to the various input patterns. Each voltage trace corresponds to stimulation by a different input vector where an active input variable is a neural ensemble of 100 neurons firing nearly synchronously in a 10 ms window. (C) Implementation of the feature binding problem using only AMPA synapses corresponding to a saturation type non-linear sub-unit and a CNF/global strategy. Top panel shows how each input makes synaptic contacts in a 10  zone either on the peri-somatic or on the distal dendritic region. Below: voltage traces in response to the various input vectors (black∶soma; blue∶distal dendrites).
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pcbi-1002867-g004: Reduced biophysical models can implement linearly non-separable functions using dendritic saturations or dendritic spikes.(A) The biophysical model. Upper panel: schematic representation of the biophysical model, where synaptic inputs are clustered (Prox and Dist), and its morphological parameters (input locations, diameters, and dendritic length are in m). Below, expected arithmetic sum versus measured somatic EPSPs: for peri-somatic AMPA stimulation (black dots) producing a linear EPSP integration; for distal NMDA stimulation (green dots) producing a spiking type non-linear summation; for distal AMPA stimulation (in blue) producing a saturation type non-linear summation. (B) Implementation of the feature binding problem using NMDA receptor synapses for the distal dendritic region, illustrating the DNF/local strategy of synaptic placement. Top panel shows how each input makes synaptic contacts in a 10 zone either on the peri-somatic or on the distal dendritic region. Below, voltage traces (black∶soma; green∶distal dendrites) in response to the various input patterns. Each voltage trace corresponds to stimulation by a different input vector where an active input variable is a neural ensemble of 100 neurons firing nearly synchronously in a 10 ms window. (C) Implementation of the feature binding problem using only AMPA synapses corresponding to a saturation type non-linear sub-unit and a CNF/global strategy. Top panel shows how each input makes synaptic contacts in a 10 zone either on the peri-somatic or on the distal dendritic region. Below: voltage traces in response to the various input vectors (black∶soma; blue∶distal dendrites).

Mentions: We implemented the FBP function in two-compartment Hodgkin-Huxley biophysical models containing a somatic and a dendritic compartment, as shown in Figure 4. The biophysical models provided a test of two assumptions made in our binary neuron model. First, we assumed that the sub-units summed their inputs independently, but in a real neuron the different sections of the neuron are electrically coupled. Here, a high intracellular resistance, and a small dendritic diameter, guaranteed a sufficient independence of the integration sites as discussed in [10]. Second, in the binary model we used discontinuous or peaked activation functions to model non-linear summation, but in neurons non-linear summation is a smooth function of the inputs. Consequently, in the biophysical model we used time-dependent AMPA conductances to generate saturations as shown on Figure 4A similar to the experimental result presented in Figure 1B. We also used NMDA time-dependent and voltage-gated conductances to generate dendritic spikes as shown on Figure 4A similar to the experimental result presented in Figure 1A. Consequently, our proof of principle model demonstrated not only how a neuron might implement linearly non-separable Boolean functions, but also how it can do so when moderately relaxing two assumptions: complete independence of sub-units and sharp activation functions.


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

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

Reduced biophysical models can implement linearly non-separable functions using dendritic saturations or dendritic spikes.(A) The biophysical model. Upper panel: schematic representation of the biophysical model, where synaptic inputs are clustered (Prox and Dist), and its morphological parameters (input locations, diameters, and dendritic length are in m). Below, expected arithmetic sum versus measured somatic EPSPs: for peri-somatic AMPA stimulation (black dots) producing a linear EPSP integration; for distal NMDA stimulation (green dots) producing a spiking type non-linear summation; for distal AMPA stimulation (in blue) producing a saturation type non-linear summation. (B) Implementation of the feature binding problem using NMDA receptor synapses for the distal dendritic region, illustrating the DNF/local strategy of synaptic placement. Top panel shows how each input makes synaptic contacts in a 10  zone either on the peri-somatic or on the distal dendritic region. Below, voltage traces (black∶soma; green∶distal dendrites) in response to the various input patterns. Each voltage trace corresponds to stimulation by a different input vector where an active input variable is a neural ensemble of 100 neurons firing nearly synchronously in a 10 ms window. (C) Implementation of the feature binding problem using only AMPA synapses corresponding to a saturation type non-linear sub-unit and a CNF/global strategy. Top panel shows how each input makes synaptic contacts in a 10  zone either on the peri-somatic or on the distal dendritic region. Below: voltage traces in response to the various input vectors (black∶soma; blue∶distal dendrites).
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pcbi-1002867-g004: Reduced biophysical models can implement linearly non-separable functions using dendritic saturations or dendritic spikes.(A) The biophysical model. Upper panel: schematic representation of the biophysical model, where synaptic inputs are clustered (Prox and Dist), and its morphological parameters (input locations, diameters, and dendritic length are in m). Below, expected arithmetic sum versus measured somatic EPSPs: for peri-somatic AMPA stimulation (black dots) producing a linear EPSP integration; for distal NMDA stimulation (green dots) producing a spiking type non-linear summation; for distal AMPA stimulation (in blue) producing a saturation type non-linear summation. (B) Implementation of the feature binding problem using NMDA receptor synapses for the distal dendritic region, illustrating the DNF/local strategy of synaptic placement. Top panel shows how each input makes synaptic contacts in a 10 zone either on the peri-somatic or on the distal dendritic region. Below, voltage traces (black∶soma; green∶distal dendrites) in response to the various input patterns. Each voltage trace corresponds to stimulation by a different input vector where an active input variable is a neural ensemble of 100 neurons firing nearly synchronously in a 10 ms window. (C) Implementation of the feature binding problem using only AMPA synapses corresponding to a saturation type non-linear sub-unit and a CNF/global strategy. Top panel shows how each input makes synaptic contacts in a 10 zone either on the peri-somatic or on the distal dendritic region. Below: voltage traces in response to the various input vectors (black∶soma; blue∶distal dendrites).
Mentions: We implemented the FBP function in two-compartment Hodgkin-Huxley biophysical models containing a somatic and a dendritic compartment, as shown in Figure 4. The biophysical models provided a test of two assumptions made in our binary neuron model. First, we assumed that the sub-units summed their inputs independently, but in a real neuron the different sections of the neuron are electrically coupled. Here, a high intracellular resistance, and a small dendritic diameter, guaranteed a sufficient independence of the integration sites as discussed in [10]. Second, in the binary model we used discontinuous or peaked activation functions to model non-linear summation, but in neurons non-linear summation is a smooth function of the inputs. Consequently, in the biophysical model we used time-dependent AMPA conductances to generate saturations as shown on Figure 4A similar to the experimental result presented in Figure 1B. We also used NMDA time-dependent and voltage-gated conductances to generate dendritic spikes as shown on Figure 4A similar to the experimental result presented in Figure 1A. Consequently, our proof of principle model demonstrated not only how a neuron might implement linearly non-separable Boolean functions, but also how it can do so when moderately relaxing two assumptions: complete independence of sub-units and sharp activation functions.

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