Figure 1.
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.
Figure 2.
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.
Table 1.
The integer-valued parameter range used in our parameter searches depending on the neuron model [lin; sat; spk], for at most input binary variables.
Table 2.
The partial truth tables for the three linearly non-separable Boolean functions of variables.
Figure 3.
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.
Figure 4.
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).
Figure 5.
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.
Table 3.
The parameters for the biophysical models.