Any neuron can perform linearly non-separable computations

An integrate and fire neuron with dendrites (the DIF)

We started from a leaky integrate and fire (LIF). This model has a membrane V modelled by the following equation:

With τ = 20 ms the neuron time constant, v(t) the membrane voltage at time t and v_E = −62 mV which sets the resting membrane voltage. R = 20 MΩ is the value of the resistance and I_s(t) models the time varying synaptic inputs conductance.

This current depends on the difference between v(t) the neuron voltage, equal everywhere, and E_s the synaptic reversal potential (0 mV) while $g_{d^i}$ is the synaptic conductance in dendrite i. Each $g_{d^i}$ is bounded between 0 and 10pS. Each $g_{d^i}$ jumps up instantaneously to its maximal value for each incoming input spike and decays exponentially with time constant τ_s = 1 ms. In a LIF all synaptic inputs are gathered into a single umbrella and i = 1. In the present study, we introduce the Dendrited Integrate and Fire (DIF) which includes at least two dendrites (i = 2). We cluster synaptic inputs into two groups, each targeting a dendrite (one green and one blue, see Figure 1). We used the Brian software version 2 to carry out our simulations, the code is freely available on the git repository attached with this report.¹⁰

Figure 1. A dendrited integrate and fire implementing a linearly non-separable computation.

(A) A leaky integrate and fire (LIF) with two dendrites making it a dendrited integrate and fire (DIF), each half of the 4 synaptic inputs targets a distinct dendrite where g locally saturates at 10pS (B) Four stimulation episodes, filled circles stand for a >50 Hz input spike train while empty circles stand for >50 Hz input spike train. Below, we plotted the response of the DIF (black) and a LIF (grey) during the episode. We purposely removed the ticks label as the frequencies depend on the parameter of the model and input regularity. The parameters of the model can vary largely without affecting the observation. (C) Some voltage response during the 3rd episode. In the clustered case (grey), the neuron reach spike threshold three times whereas it reaches spike thresold seven times in the scattered case (black).⁵

Boolean algebra refresher

First, let’s present Boolean functions:

Definition 1. A Boolean function of n variables is a function on {0,1}ⁿ into {0,1}, where n is a positive integer.

Importantly, we commonly assume that neurons can only implement linearly separable computations:

Definition 2. f is a linearly separable computation of n variables if and only if there exists at least one vector w ∈ ${\mathcal{R}}^n$ and a threshold Θ ∈ $\mathcal{R}$ such that:

where X ∈{0,1}ⁿ is the vector notation for the Boolean input variables.

The compact feature binding problem (cFBP)

In this section, we demonstrate a class of compact linearly inseparable (non-separable) computations that we are going to study. These computations are compact because they have four input/output lines.

We entirely specify an example in Table 1. This computation that we call the compact feature binding problem (cFBP) is linearly inseparable.

Proposition 1. The cFBP is linearly inseparable (non-separable)

Proof. The output must be 0 for two disjoint couples (1,2) and (3,4) of active inputs. It means that w₁ + w₂ ≤ Θ, and w₃ + w₄ ≤ Θ, and we can add these two inequalities to obtain w₁ + w₂ + w₃ + w₄ ≤ 2Θ. However, the output must be 1 for two other couples made of the same active inputs (1,3) and (2,4). It means that w₁ + w₃ > Θ, and w₂ + w₄ > Θ, and we can add these two inequalities to obtain w₁ + w₂ + w₃ + w₄ > 2Θ. This yield a contradiction proving that no weights set exists solving this set of inequalities.

The cFBP is simple in two ways:

We can extend the cFBP by increasing the number of inputs. In this case we deal with tuples instead of couples. As such, the cFBP corresponds to an entire family of linearly inseparable computations, and a dendrited neuron can implement them using the strategy that we will present in the next section.

A LIF with its linear integration cannot implement such a computation. While a neuron with two saturating dendrites can easily implement it. We already proved how a ball-and-stick biophysical model can implement this computation in a previous study.³

Implementing the cFBP in a dendrited integrate and fire

We use two independently saturating conductances to implement the cFBP in a minimal extension of the LIF that we called the dendrited integrated and fire (DIF). The DIF has a single membrane voltage to account for its compactness so we might wonder how local saturation can arise in such a morphology. Saturation has two possible origins: (1) a reduction in driving force can cause saturation as in,¹ but (2) it can also be due to the intrinsic limitations in conductance per unit of surface. This latter possibility makes saturation possible in an electrically compact neuron. Even in a neuron with a small dendritic tree, the conductance is going to reach an upper bound per unit of surface and the only possibility to increase excitation consists in stimulating a larger area. We are going to employ this local bounding of the conductance to implement the cFBP in a DIF.

To do that, we only need two dendrites as shown in Figure 1A. We can interpret the 0s and 1s in the truth table in at least two ways: (1) either the pre- or post-synaptic neurons activates (2) or they reach a given spike frequency. In the following section, we will use the latter interpretation. Consequently, we consider a pre-synaptic input active when it fires above 50 Hz regular spike-train and inactive if it fires below 50 Hz (this value is arbitrary and can largely vary to match a neuron working range). We stimulate our model in four distinct episodes to reproduce the truth table from the previous section. You can observe on Figure 1 that locally bounding g enables implement the of cFBP. When g has no bound, the membrane voltage always reaches the spiking threshold at the same speed (LIF case). When we locally bound conductances the membrane voltage takes more time (45 ms see Figure 1C) to reach threshold in the clustered case (total g = 10pS) than in the scattered case (total g = 20pS). All in all, a DIF will respond differently for the clustered and scattered case while a LIF won’t. This enables a DIF to implement the cFBP while a LIF can’t.