Neurons with dendrites can perform linearly separable computations with low resolution synaptic weights

Biophysical neuron model

We performed simulations in a spatially extended neuron model, consisting of a spherical soma (diameter 10 µm) and two cylindrical dendrites (length 400 µm and diameter 0.4 µm). The two dendrites are each divided into four compartments and connect to the soma at one extremity.

In contrast to a point-neuron model, each compartment has a distinct membrane potential.

The membrane potential dynamics of the somatic compartment follows the Hodgkin-Huxley formalism with:

The dendritic compartments only contain passive currents:

Here, V_soma and V_dend are the respective membrane potentials, C_m = 1µFcm⁻² is the membrane capacitance, g_L, $\overline{g}$_K, and $\overline{g}$_Na stand for the leak, the maximum potassium and sodium conductances respectively, and E_L, E_K, and E_Na stand for the corresponding reversal potentials. The currents I_a represent the axial currents due to the membrane potential difference between connected compartments. The synaptic current I_s arises from a synapse placed at the respective compartment. It is described by

with E_s being the synaptic reversal potential and g_s the synaptic conductance. This conductance jumps up instantaneously for each incoming spike and decays exponentially with time constant τ_s = 1 ms otherwise:

The dynamics of the gating variables n, m, and h are identical to 14, except for shifting the membrane potential relative to V_T = –50 mV instead of the cell’s resting potential. The equations are omitted here for brevity. The parameter values are summarized in Table 1. Note that due to the absence of sodium and potassium channels in the dendrites, the dendrites are passive and cannot generate action potentials.

All simulations were performed with Brian 2¹⁵. The code is available at http://doi.org/10.5281/zenodo.4315011¹⁶. It allows for reproducing the results presented in Figure 4, Figure 5 and Figure 6. To demonstrate that the details of the neuron model do not matter for the results presented here, the provided code can also be run with a simpler leaky integrate-and-fire model.

Elementary neuron model and Boolean functions

As a reminder, we first define 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.

Note that we use the terms function and computation interchangeably.

A special class of Boolean functions, which are of particular relevance for neurons, are 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 ∈ ℝⁿ and a threshold Θ ∈ ℝ such that:

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

Binary neurons are one of the simplest possible neuron models and closely related to the functions described above: their inputs are binary variables, representing the activity of their input pathways, and their output is a single binary variable, representing whether the neuron is active or not. The standard model is a linear threshold unit (LTU), defined as follows:

Definition 3. An LTU has a set of m weights w_i ∈ 𝒲 and a threshold Θ ∈ 𝒯so that:

where X = (X₁, . . . , X_m) are the binary inputs to the neuron, and 𝒲 and 𝒯 are the possible values for synaptic weights and the threshold, respectively.

This definition is virtually identical to Definition 2, however, w_i and Θ are no longer arbitrary real values, but chosen from a finite set of numbers depending on the specific implementation and noise at which these value can be stabilised. It follows that a neuron may not be able to implement all linearly separable functions. For instance, a neuron with non-negative weights can only perform positive linearly separable computations:

Definition 4. A threshold function f is positive if and only if f (X) ≥ f (Z) ∀(X , Z) ∈ {0, 1}ⁿ such that X ≥ Z (meaning that ∀i: x_i ≥ z_i).

To account for saturation occurring in dendrites, we introduce the sub-linear threshold unit (SLTU):

Definition 5. An SLTU with d dendrites and n inputs has a set of d × n weights w_i,j ∈ {0, 1} with n w_i such that ${\displaystyle {\sum }_{j=1}^d{w}_{i,j}=w_i}$ and a threshold Θ ∈ 𝒯, such that:

with

The function E accounts for dendritic saturation; because we work with binary weights its value is either 0 or 1.

Such a neuron model can implement all positive Boolean computations (see Definition 4) given a sufficient number of dendrites and synapses¹¹.

We used integer-valued and non-negative parameters both for the LTU and the SLTU without loss of generality. It allows us to exactly determine the minimal resources necessary to implement a given computation.

Implementation of computations with three input variables

We begin by listing all computations of n = 3 inputs that are implementable by an LTU (i.e., positive threshold functions; Table 2). These computations can be divided in five classes, and one can obtain all computations from a class by swapping the input labels. The OR, AND/OR, and AND can be implemented with equal synaptic weights. In contrast, the remaining classes require heterogeneous synaptic weights. We call these classes the Dominant AND (D-AND) and the Dominant OR (D-OR): to implement these computations, an LTU needs to have one synaptic weight that is twice as big as the others (see Figure 1).

The D-AND computation gets its name from the fact that it requires the activation of a dominant (D) input AND the activation of another input. The D-OR is the Boolean dual of the D-AND, i.e. obtained by replacing AND operations by OR, and vice versa. In this computation, activation of the dominant input OR of the two other inputs together triggers an output. Both computations have a “dominant input” – an input that is sufficient to make the output true (D-OR), respectively necessary to make the output true (D-AND). There is nothing comparable in the other three computations, which treat all inputs identically. In the present paper, we always chose X₁ as the dominant input, but we could have picked X₂ or X₃.

Figure 1. Minimal implementation of the Dominant AND computation (D-AND) and its dual by a linear threshold unit (LTU).

Implementations of the D-AND where X₁ is the dominant input. Squares represent synapses with their synaptic weight, and circles stand for transfer functions. Here, the transfer functions are threshold functions with the given value as their threshold. A: Implementation of the D-AND, note that X₁ has twice the synaptic weight compared to the others. B: Implementation of the D-OR, note that we keep the same synaptic architecture and we only change the threshold of the transfer function.

An LTU (Figure 1) implements D-AND and D-OR by making use of synaptic strength to distinguish between the dominant and non-dominant inputs. We employed synaptic weights with integer values to reflect their finite precision. Even if synaptic weights can take real values, a finite precision means a finite number of values, which again can be represented by an integer value. The weight and threshold values to implement a function are obviously not unique. For example, we could multiply all the weights by 2 and set the threshold to 6 (D-AND), or 4 (D-OR) and obtain the same results. Here, we always use the lowest possible integer values for synaptic weights, and the corresponding lowest possible threshold.

Next, we wanted to implement the D-AND and D-OR computation in threshold units with non-linear dendritic sub-units, as an abstraction of neurons with dendrites⁷.

We consider two types of non-linearities: a threshold function to model supra-linear summation; and a saturating function to model sub-linear summation (SLTU; see Methods). Both types of summation have been observed in dendrites. Dendritic spikes are a well-known example of supra-linear summation¹², while sub-linear summation can be observed in completely passive dendrites due to a reduced driving force⁹.

On the one hand, Figure 2 (top) shows that a neuron with supra-linear dendrites implements the D-OR using space whereas the sub-linear implementation uses strength. On the other hand, Figure 2 (bottom) shows that a neuron with supra-linear dendrites implements the D-AND using strength whereas the sub-linear implementation uses space.

In both cases, all synapses are of identical strength. However, note that in the supra-linear implementation of the D-AND in Figure 2C the X₁ input connects to both dendrites. Therefore, if we define an input’s synaptic weight as the total effect it has in the final summation stage (analogous to depolarisation measured in the soma of a neuron), we have to consider the weight of X₁ as twice as high as the other inputs. This makes this implementation “as bad as” the implementation in an LTU (Figure 1A): the dominance of X₁ is expressed by a stronger weight.

This starkly contrasts with the sub-linear implementation of the D-AND (Figure 2D), where all synaptic weights are identical. The placement of X₁’s synapse causes its dominance: while X₂ and X₃ share a dendrite, X₁’s synapse lays alone on a dendrite. This implementation uses space. We focus on sub-linear summation and the D-AND for the rest of the study.

Figure 2. Minimal implementation of the Dominant AND computation (D-AND) and its dual (D-OR) by a neuron with dendrites.

Squares represent synapses and circles represent transfer functions with their respective threshold/saturation values. Note that the final transfer functions (“somatic integration”) are always threshold units, whereas the transfer functions of the sub-units (“dendrites”) are threshold functions for supra-linear summation, and saturating functions (corresponding to the E function defined in Definition 5) for strictly sub-linear summation. A: D-AND implementation using sub-linear summation where X₁ targets only one dendrite. B: D-OR implementation, in this case X₁ targets two sub-linear dendrites. C: D-AND implementation using supra-linear summation, where X₁ targets two dendrites. D: D-OR implementation, X₁ in this case targets only one dendrite.

Implementing the D-AND for an arbitrary number of input variables

In the previous section, we have limited our analysis to computations with three input variables. We will now extend the definition of the D-AND to an arbitrary number of input variables. As in the three-variables case, we will consider one input to be the dominant input (assumed to be X₁, without loss of generality). This input has to be activated together with at least one of the non-dominant inputs. Formally, we therefore define f_n(X) as follows:

where X is the n-dimensional input vector with elements X₁... X_n.

We can implement this computation in an LTU (Figure 3A), as well as in an SLTU (Figure 3B)

In the LTU implementation (Figure 3A), the D-AND of n variables requires that an input has a synaptic weight at least n − 1 times bigger than the other inputs, and the threshold has to grow accordingly.

We can summarise these observations in a proposition.

Proposition 1. To implement the D-AND, an LTU requires that an input has a synaptic weight n − 1 times bigger than the smallest synaptic weight.

Proof. The LTU must stay silent when X₁ is not active, even if X₂, X₃, . . . , X_n are active. Therefore w₂ + w₃ + ... + w_n < Θ, thus Θ must be at least n × w_min with w_min the smallest synaptic weight.

Conversely, the output should be active as soon as X₁ is co-active with any other input X_j (for j > 1). So w₁ + w_min ≥ Θ, this means w₁ + w_min ≥ n × w_min, thus w₁ ≥ w_min(n − 1).

In contrast, Figure 3B provides a constructive proof that an SLTU can implement the D-AND with equal synaptic weights. In this implementation, the distinguishing feature of the dominant input is that it targets the second dendrite; synaptic weights and the threshold do not have to change with the number of inputs. If one only measured the response to single inputs at the “soma” (last stage of summation), the dominant input would be indistinguishable from the other inputs, despite its dramatically different importance.

Figure 3. Extending the D-AND implementation to n inputs.

Synaptic weights are in squares, and transfer functions are in circles. A: Minimal D-AND implementation in an LTU. Note that this implementation requires a synaptic weight that is n − 1 times bigger than the smallest weight. B: Implementation in an SLTU with sub-linear summation (see Definition 5).

We will see next how these insights transfer to a more realistic biophysical model.

Implementation of the D-AND in a biophysical model

Figure 4A presents a biophysical model of a single neuron implementing the D-AND computation with three groups of synapses. All the synapses, taken individually, produce the exact same depolarisation at the soma because we place them at the same distance (350 µm) and give them the same maximal conductance (20 nS).

We first look at the sub-threshold behaviour by disabling the sodium channels in the soma ($g_{Na}^{max}$ = 0). Figure 4B plots the somatic voltage response at distinct locations in response to either clustered (black) or dispersed (aquamarine) synaptic activation. Despite activating the same number of synapses in both cases, and despite them all having the same strength, the depolarisation is markedly different. When we disperse active synapses, EPSPs sum linearly (same as dotted gray line) whereas when we cluster active synapses summation becomes sub-linear. This difference is robust with respect to the specific values of the synaptic weights. As shown in Figure 4C, the dispersed activation always exceeds the clustered activation, for the same total synaptic weight. This difference remains even for a total weight bigger for the clustered than the dispersed case. For example, a clustered activation with a total weight of 100 nS leads to a maximum membrane potential of only −54mV in the soma, whereas a dispersed activation with a mere total weight of 10 nS leads to a maximal membrane potential of −52.5mV.

Figure 4. A biophysical model sensitive to synapses’ spatial distribution.

A: A biophysical model with two dendrites and a soma (lines: dendrites, circle: soma). Coloured squares depict synapses. The model has three equivalent groups of synapses (black/blue/green). B: Somatic membrane voltage traces in response to synaptic inputs which are either clustered (synapses activate on the same dendrite; aquamarine) or dispersed (synapses activate on distinct dendrites; black). In a linear neuron, all active groups of synapses (black + blue or black + green or blue + green) produce the same somatic EPSP (gray dotted line). C: Maximal membrane voltage at the soma depending on the total synaptic weight for either clustered (aquamarine) or dispersed (black) stimulation.

We can explain this observation by considering the synaptic driving force¹⁷. The synaptic current induced by the activation of the synapse depends on the distance between the membrane potential and the synapses’ reversal potential; when several inputs drive the membrane potential closer to the reversal potential (here 0mV), this driving force diminishes. The combined effect of multiple synaptic inputs is therefore smaller than what is expected from summing the individual effects. In other words, the dendrite performs sub-linear summation.

This means that even if we have a complete synaptic democracy¹⁸ (all synapses have the same impact on the soma when taken individually), the relative placement of the synapses strongly influences the somatic response.

Based on the sub-threshold behaviour presented above, we will now show that we can implement the D-AND in a spiking neuron model. It is crucial to look at the supra-threshold behaviour as it is how the neuron communicates with the rest of the network. Moreover, backpropagated action potentials might undermine the dendritic non-linearity disrupting the implementation¹⁹.

We can interpret Boolean inputs and outputs in different ways when we apply them to a biophysical spiking neuron model. Here, we will consider two interpretations. Firstly, we can think of an active input as corresponding to a continuous stimulation where the individual spikes arrive at random times, and of an active output as some spiking activity of the neuron (“rate interpretation”). Alternatively, we can think of active inputs as coincidentally arriving spikes within a certain time window, and accordingly of an active output as a single spike emitted in response (“spike interpretation”). We present the model implementing the rate interpretation in Figure 5. We introduced this model earlier (Figure 4), except that it now has active sodium channels in the soma ($g_{Na}^{max}$ = 650mScm^–2). Each of its inputs (colours corresponding to the colours in Figure 4) activates in 25 randomly chosen time-bins of 1 ms to simulate a 100 Hz spike train over 250 ms.

The Figure 5 displays, from top to bottom, the model’s responses in five different situations:

Figure 5. A biophysical model implementing the Dominant AND (rate interpretation).

In this interpretation X_i =1 corresponds to a presynaptic neuron firing at 100Hz. Top: activity of the three input synapses, the two first synapses impinge on the same dendrite while the black one impinges on another. Bottom: Eight somatic membrane responses depending on the active inputs. (gray: no synapse/only black/green/blue, green: black + green, blue: black + blue, aquamarine: green + blue, black: all inputs active). Note that this activity reproduces the truth table from Table 2. The difference between the aquamarine line (green and blue inputs) and the green and blue lines (black input and either green or blue input) is due to the sub-linear summation in the dendrite. With linear summation these three input patterns would evoke identical responses.

This figure thus presents the response of the neuron model to all non-trivial cases, we have only omitted the case without any input activation (and therefore without any output activity).

Figure 6. A biophysical model implementing the Dominant AND (spike interpretation)

Top: The biophysical model receives input from three sources, where activation happens at regular intervals of 25 ms, with a random jitter of ±1ms for each spike. We translate this activity into a binary pattern for each time bin of 25 ms. Bottom: The model’s membrane potential as measured in the soma. The response spikes implement the output of the D-AND computation as described in Table 2.

Finally, we show an implementation of the spike interpretation in Figure 6. This model is identical to the model shown previously (Figure 5), except for a slightly lower activation threshold of the sodium channels (V_T = −55 mV instead of V_T = −50 mV) to make it spike more easily. We discretize time into bins of 25 ms and decide randomly for each input whether it is active in each bin. If it is active, it activates at the beginning of the bin with a small temporal jitter (1 ms); inputs activating in the same bin therefore spike coincidentally. We can directly link these activations to Boolean variables that are either 0 (no spike) or 1 (spike). As Figure 6 shows, the neuron implements the D-AND and only spikes whenever the black synapses activate together with at least one of the blue or green synapses.

We have shown that a biophysical model can implement the D-AND computation using a different strategy than the LTU. Each input has the same synaptic weight producing the same depolarisation at the soma. To distinguish between the inputs, the biophysical model uses location instead of strength: the dominant input (black) targets its own dendrite, while the two other inputs cluster on the same dendrite. With this strategy, the model can implement the D-AND. This implementation also works for two interpretations of the Boolean inputs and outputs – as elevated rates of spiking without temporal alignment, or as precisely timed coincident spikes.