Figure 4.

Using Boolean algebra to analyze binary neuron models with dendritic nonlinearities. (A) Truth tables for the Boolean functions AND, OR and XOR for two synaptic inputs (x₁ and x₂). The two colored horizontal lines illustrate how the AND and OR functions are linearly separable, i.e., a single line divides all inputs between two groups, one group having an output of 0 and the other group having an output of 1. Neuron output binary value is denoted as y. (B) Threshold linear unit model neuron with two inputs. The weight of each input is represented by the area of the black disc drawn between the input and the model neuron. Here all weights are equal to 1. A spiking threshold (θ) of 2 allows the model neuron to compute the AND function (left), whereas if θ =1 the neuron computes the OR function (right). (C) Top, Simplified representations of a supralinear sI/O (left) and its mathematical approximation by a Heaviside function (right) with a height h and a threshold θ. Bottom, simplified representation of a sublinear sI/O and its mathematical approximation by a piecewise linear, then saturating function. (D) Generalized diagram representing a two-layer integration model neuron with several compartments and n inputs. Each branch represents a dendritic compartment, and the integration operation performed by this compartment is represented by the box on the branch. The threshold θ and the output value h of the nonlinearity are indicated within the box. The result from the integration from each branch is then linearly summed and compared to the somatic spike threshold Θ. (E) Implementation of the (partial) feature binding problem (pFBP) by binary neurons with two dendritic compartments D₁ and D₂, either supralinear or sublinear. Top, truth table describing various input feature combinations, the response of each dendritic compartment, D (0:inactive/1:active), and the final neuronal output, y. Columns with green shading are the outputs of dendrites exhibiting supralinear operations, while columns shaded in blue contain outputs of dendrites that exhibit sublinear operations. Bottom, Model neuron with equivalent dendrite representation that can implement the pFBP using supralinear (left) or sublinear dendritic compartments (right), with θ and h values indicated in the box. If dendritic integration is supralinear, two groups of inputs are needed to activate a compartment, and a single compartment can trigger a spike. If dendritic integration is sublinear, a single input can activate the dendritic compartment and the two compartments must be active to trigger a spike.