Contribution of sublinear and supralinear dendritic integration to neuronal computations

Abstract

Nonlinear dendritic integration is thought to increase the computational ability of neurons. Most studies focus on how supralinear summation of excitatory synaptic responses arising from clustered inputs within single dendrites result in the enhancement of neuronal firing, enabling simple computations such as feature detection. Recent reports have shown that sublinear summation is also a prominent dendritic operation, extending the range of subthreshold input-output (sI/O) transformations conferred by dendrites. Like supralinear operations, sublinear dendritic operations also increase the repertoire of neuronal computations, but feature extraction requires different synaptic connectivity strategies for each of these operations. In this article we will review the experimental and theoretical findings describing the biophysical determinants of the three primary classes of dendritic operations: linear, sublinear, and supralinear. We then review a Boolean algebra-based analysis of simplified neuron models, which provides insight into how dendritic operations influence neuronal computations. We highlight how neuronal computations are critically dependent on the interplay of dendritic properties (morphology and voltage-gated channel expression), spiking threshold and distribution of synaptic inputs carrying particular sensory features. Finally, we describe how global (scattered) and local (clustered) integration strategies permit the implementation of similar classes of computations, one example being the object feature binding problem.

Keywords: Boolean analysis; binary neruons; dendrites; input-output transformation; neural computation; nonlinear transformations; uncaging; votlage activated channels.

Figure 1

Dendritic operations and their influence on neuronal firing. (A) Schematic diagram of a subthreshold synaptic input-output experiment in a neuron with supralinear dendritic compartments (left, supralinear compartments in green, linear compartments in black) or in a neuron with sublinear dendritic compartments (right, sublinear compartments in blue). The red spots are sites of synaptic activation or sites of glutamate uncaging. (B) Somatic voltage responses evoked by simultaneous synaptic activation or uncaging. Green curves are responses evoked with increasing number of synapses activated within a supralinear dendrites. Blue traces are similarly obtained within a sublinear dendrite. (C) Arithmetic sum of individual responses to synaptic activation or uncaging. (D) Subthreshold input/output relationships (sI/O) used to quantify dendritic operations. The dashed line represents a linear releationship. Two horizontal dotted lines indicate two example somatic spike thresholds (θ₁ and θ₂). (E,F) Example of synaptic integration of three synaptic inputs distributed across the dendritic tree (E) or clustered on a single dendritic branch (F) of a neuron with supralinear dendritic compartments (left) or sublinear compartments (right). The output spike train, and hence neuronal computation, differs depending on the threshold. The more depolarized threshold value (θ₁) allows the neuron with supralinear dendrites to exhibit a cluster-sensitive neuronal computation (fires only when three inputs are activated in the same compartment). The θ₁ threshold also allows a neuron with sublinear dendrites to exhibit scatter-sensitive neuronal computations. The lower threshold (θ₂) imparts a different neuronal computation based on simple linear summation and is not sensitive to activated synapse location.

Figure 2

Theoretical basis for sublinear summation within passive dendrites. (A) Equation (1) describes the different current components underlying an EPSP in a single electrical compartment. Integration of this equation describes the variation of the membrane voltage over time. The transient change in driving force (ΔV = V_m − E_syn) is determined by the amplitude and time course of the local dendritic EPSP (black trace). At the peak of the EPSP (solid blue arrow) the driving force is maximally reduced, and then recovers back to that at resting membrane potentials during the EPSP decay (dotted blue arrow). The reduced driving force decreases the synaptic current, and hence the net depolarization, creating a sublinear relationship between EPSP and its underlying conductance. (B) Equivalent circuit for dendritic cables, where g_m and c_m are the membrane conductance and capacitance, respectively, and r_a is the axial resistance of a unit of cable. A synapse is represented in the circuit (by the synaptic conductance G_syn and the synaptic reversal potential E_syn). For an infinite cable, the spatio-temporal distribution of voltage is described by the relation (2), where τ_m is the membrane time constant, and λ is the length constant. The length constant relationships are derived from solving the cable equation (2) for step changes in membrane voltage (λ_DC) or for a sinusoidal membrane potential change (λ_AC). The latter is helpful to understand the dendritic filtering of transient EPSPs. Equation (3) is the relation for the input resistance R_D for an infinite cable. (C) Top, ball-and-stick model of a neuron with colored arrows indicating the location of three synapses (Syn 1–3). The graph above the diagram represents the peak amplitude of a dendritic EPSP as a function of distance. Bottom, the two graphs describe respectively the dendritic and somatic depolarizations in response to individual (colored lines) or combined synaptic inputs (black lines). Concomitant activation of two neighboring synaptic inputs (within ~λ_AC) will therefore mutually reduce their driving force and sum sublinearly (for example synapses 1 and 2, solid black trace for the EPSP observed in response to their simultaneous activation, dashed black trace for the arithmetic sum of the individual EPSPs). More separated synapses will, however, sum more linearly (synapses 1 and 3, gray trace).

Figure 3

Contribution of dendritic and synaptic properties to EPSP summation. (A) Influence of morphological parameters dendritic: diameter (left), increasing distance to soma (middle) and increasing dendritic branching (right) on the dendritic sI/O. The inserts illustrate the effect of morphology on somatic EPSPs under the different conditions. Synapse location and traces are color coded. Dashed line shows a linear I/O for reference. (B) The role of ion channels on the shape of the sI/O, for a given morphology. Either K⁺ channels, (orange), Na⁺ channels (green), VGCC (pink), or NMDA receptors (sky blue) are added to a passive dendrite (blue). (C) Example of sI/O in three realistic combinations: thick (>2 µm) dendrites with active conductances (blue curve, as in Branco and Häusser, 2011), thinner dendrites with active conductances (brown curve, <1 µm, Losonczy and Magee, 2006), or thin dendrites with only passive properties (blue curve, Abrahamsson et al., 2012). (D) Influence of synaptic properties on the sI/O for a given morphology and ion channel combination. An increase in synaptic strength makes the sI/O diverge from linearity both in the sublinear and the supralinear regime, whereas increasing the interval or the distance between synaptic inputs tends to linearize the curve (right).

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.

Figure 5

Computing a linearly non-separable function (full FBP) with supralinear and sublinear dendrites and using local vs. global synaptic wiring strategies. (A) Left, model neuron with equivalent dendrite representation of two compartments, linear (black) and supralinear (green), and a clustered distribution of object features (object 1 : ×1, ×2 and object 2 : ×3, x4) (local strategy). Right, schematic representations of synaptic placements equivalent to the model on the left. (B) Left, model neuron with equivalent dendrite representation of two compartments, linear (black) and sublinear (blue), and a distributed placement of inputs carrying object features. (C–E) Implementation of the full FBP (y = 1; “apple shape and red” or “banana shape and yellow”). (C), implementation of the full FBP using a model with a supralinear compartment and a local wiring strategy. Inactive inputs are represented in light gray and the corresponding feature in lighter color. (D) Implementation of the full FBP using a model with a supralinear compartment and a global wiring strategy. The area of the disc adjacent to a compartment next to each object feature represents the relative weight of this feature. Here the relative weights used are of 1 and 2. (E) Implementation of the full FBP using a model with sublinear compartment and a global wiring strategy.