All neurons can perform linearly non-separable computations

An integrate and fire neuron with interacting synapses (the SIF)

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 = −65 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 current. Each time the voltage reaches V_t = −62 mV a spike is triggered and the voltage is resetted to −65 mV. We used the following equation to account for the synaptic inputs.

The synaptic current depends on the difference between v(t) the neuron voltage, equal everywhere, and E_s the synaptic reversal potential (0 mV). In the present work, we have four input sources and contrary to what is done usually we have only two conductances g₁ and g₂ which collapse conductance from input 1,2 and 3,4 respectively. This account for the interaction between the input sources and do not consider them fully independent as it is usually the case. Each g_i is bounded between 0 and 100 pS. Each g_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 work we cluster synaptic inputs into 2 groups (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.

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

(A) A leaky integrate and fire (LIF) with two saturating points, each half of the 4 synaptic inputs targets a distinct point where g locally saturates at 100 pS (B) Four stimulation episodes, filled circles stand for 50 Hz poisson spike trains while empty circles stand for no input spike train. Below, we plotted the response of the LIF (grey) and of the SIF (black) during an episode. We purposely removed the ticks label as the frequencies depend on the parameter of the model and input correlation. The parameters of the model can vary largely without affecting the observation. (C) Somatic voltage response of the SIF when filled circle means spike and empty circle means no-spike. One can observe that the SIF reproduces the truth table also in this case.

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 only need to be defined on have four input lines. Changing the other input lines would not affect our result.

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 compact beauce it specifies only four lines of a function. A complete definition would include 16 distinct input/output relationship. This incomplete definition of the function leaving all the remaining input/output relation is the minimal. This computation is as complex as the famous exclusive OR (XOR). Note here that our SIF can also implement the XOR using a parameter set explained here [?]. However, contrary to the XOR it can be implemented with excitatory inputs and a monotone transfer function.⁸

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 SIF 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 groups of saturating synapses can easily implement it. We already proved how a ball-and-stick biophysical model can implement this computation in a previous study.⁸

Implementing the cFBP in a saturating integrate and fire (SIF)

We use two independently saturating conductances to implement the cFBP in a minimal extension of the LIF. The SIF 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 Ref. 6, 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. In every cases 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 SIF.

To do that, we only need two saturating points as shown in Figure 1A. We can interpret the 0 s and 1 s 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 two interpretations. We first consider a pre-synaptic input active when it fires a 50 Hz poisson spike-train and inactive if it does not fire (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 two interpretation of the truth table: either a rate based or a spike based interpretation. In both cases we ca observe that locally bounding g enables to implement the 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 total input current is higher if inputs target two points rather than one (total g = 100 pS Vs g = 200 pS). All in all a SIF will respond differently for the clustered and scattered case while a LIF won’t. This enables a SIF to implement the cFBP while a LIF can’t.