← SciSim / Neuroscience
· · SciSim ·

Dendritic Computation

Passive & Active Dendrites · Coincidence Detection · Nonlinear Integration

🧠 Tier: Standard Undergraduate / Graduate · Active Dendrites · NMDA Spikes · Compartmental Models
Version:
§ 01
Interactive Simulation
Var 1
—
—
Var 2
—
—
Input
—
—
Output
—
—
Count
0
—
Mode
—
—
Time
0
ms
Playback
Preset
Parameters
Parameter 15.0
Parameter 25.0
Input I5.0
T sim (ms)500
Overlays
§ 02
The Idea, Step by Step
▸ From a Single Branch to a Tiny Neural Network

Picture a single brain cell not as one tiny on/off switch, but as a little committee. The thin branches reaching out from a neuron — its dendrites — were long imagined as passive wires that just funnel signals down to the cell body. In fact, each branch can make its own small decision before passing anything on. It is like a checkpoint where two guards must both swipe their cards at the same moment before the door will open.

STEP 1 — Each branch adds up its inputs
Every synapse delivers a small voltage bump (an EPSP). If only one arrives, the branch barely stirs. But if several land on the same branch at nearly the same time, their bumps pile on top of one another. The branch is, in effect, keeping a running total of how much excitation it is receiving right now.
STEP 2 — Past a threshold, the branch "spikes" on its own
Once the local voltage climbs past a threshold (around $-45$ mV), voltage-gated NMDA channels swing open and pour in extra current — a sudden, self-reinforcing surge called a dendritic (NMDA) spike. One input alone gives a tiny response; two together can give a response many times larger than their simple sum. In symbols the branch is superlinear: $\text{resp}_{A+B} > \text{resp}_{A} + \text{resp}_{B}$. That behaviour is an AND gate — neither input alone is enough, but both together cross the line.
STEP 3 — Many branches make a two-layer network
Formally, each branch applies a nonlinear function $\sigma$ to its summed input, and the soma then sums the branch outputs and applies its own threshold: $V_{\text{soma}} \approx \sigma\!\big(\sum_j w_j\,\sigma(\sum_i w_{ij}x_i)\big)$. That two-stage shape is precisely a two-layer neural network, with every dendritic branch acting as one hidden unit. It is why a single pyramidal neuron with active dendrites can separate input patterns — even compute XOR — that a single "point" neuron never could. In the simulator, the Input I slider sets how hard the synapses drive a branch, while Parameter 1 and Parameter 2 set the branch coupling and the spike threshold.

Try this in the sim above: (1) Turn Input I up while the branch is only weakly coupled and watch it stay quiet — below threshold, no dendritic spike. (2) Push the drive past the threshold and see the output jump by far more than double — the NMDA spike has fired. (3) Lower the threshold parameter and notice the strict AND behaviour loosen toward an easier-to-trigger, OR-like response.

§ 03
Equation Derivation
▸ Passive vs Active Dendritic Integration

Passive dendrites act as linear integrators — synaptic inputs sum linearly at the soma. Active dendrites (with voltage-gated channels) can implement nonlinear operations: AND gates, multipliers, and local coincidence detection.

$$\text{Passive (linear):}\quad V_{\text{soma}} = \sum_i w_i \cdot g_{\text{syn}_i} \cdot e^{-L_i} \qquad \text{(superposition principle)}$$ $$\text{Active (nonlinear):}\quad V_{\text{soma}} = f\!\left(\sum_i w_i \cdot g_{\text{syn}_i}\right) \qquad f = \text{sigmoidal (NMDA spike)}$$

The dendritic nonlinearity threshold for NMDA spike initiation:

$$V_{\text{dend}} > V_{\text{NMDA\_thresh}} \;\Rightarrow\; \text{NMDA spike: } V_{\text{dend}} \text{ jumps to plateau potential (+20 mV locally)}$$

Two-layer computation model (Poirazi & Mel 2001):

$$V_{\text{soma}} \approx \sigma\!\left(\sum_j w_j \cdot \sigma\!\left(\sum_{i \in \text{branch}_j} w_{ij} x_i\right)\right)$$
▸ Derivation Steps
STEP 1 — Two-Compartment Model
The simplest active dendrite model: soma (s) + dendrite (d) connected by axial resistance R_ax. C_d dV_d/dt = −g_L(V_d−E_L) + g_NMDA(V_d)(V_d−0) + I_syn − (V_d−V_s)/R_ax. C_s dV_s/dt = −g_L(V_s−E_L) − g_Na m(V_s)(V_s−E_Na) − g_K n(V_s)(V_s−E_K) + (V_d−V_s)/R_ax. The soma generates APs; the dendrite generates local dendritic spikes if enough synaptic input arrives to trigger NMDA channel opening.
STEP 2 — NMDA Spike: All-or-Nothing in a Dendrite
When multiple co-active synapses depolarise a dendritic branch above ~−40 mV, NMDA receptors in that branch are partially unblocked. This provides positive feedback: more depolarisation → more Mg²⁺ block relief → more NMDA current → more depolarisation. This bistability creates a local 'dendritic spike': V_dend rapidly jumps to a plateau potential (~0 to +10 mV) and remains there as long as synaptic input persists. The NMDA spike propagates with higher amplitude and less attenuation than a passive EPSP, dramatically amplifying distal synaptic input at the soma.
STEP 3 — AND Gate Implementation
Two synapses A and B on the same dendritic branch: synapse A alone cannot trigger NMDA spike; synapse B alone cannot trigger NMDA spike; BOTH A and B active together trigger NMDA spike. This is a logical AND gate — the synapse strength w must be chosen such that w_A + w_B > V_NMDA_thresh > w_A or w_B. Poirazi & Mel (2001) showed that a single L5 pyramidal neuron with active dendrites can classify inputs with nonlinear separability equivalent to a 2-layer MLP with 20+ hidden units.
STEP 4 — Back-Propagating APs and STDP
When a somatic AP is generated, it back-propagates into the dendrites (back-propagating action potential, bAP). The bAP reaches distal dendrites with attenuated amplitude (~20–40 mV from the soma). When a bAP coincides with a synaptic input (EPSP) in a distal dendrite, the combined depolarisation fully relieves Mg²⁺ block → calcium influx via NMDA → LTP at that specific synapse. This is the Hebbian STDP mechanism implemented at the dendritic level: the bAP signals 'postsynaptic firing' to all dendrites simultaneously, enabling specific strengthening of the recently active synapses.
STEP 5 — Plateau Potentials and Up States
In L5 pyramidal neurons, strong dendritic input can trigger a dendritic Ca²⁺ plateau potential lasting 100–500 ms — much longer than a single AP. These plateaus drive sustained somatic firing during the plateau. In vivo, plateau potentials are thought to generate the 'UP states' seen during slow-wave sleep and anaesthesia: the dendrite maintains the membrane in a depolarised, high-conductance state via persistent Ca²⁺ and NMDA currents. This mechanism has been proposed as the cellular basis of working memory maintenance — a single cell can 'latch' into a persistent active state via a dendritic plateau.

▸ Worked Example — Two-Input AND Gate

Dendritic branch with two synapses. NMDA spike threshold: V_NMDA = −45 mV. Passive attenuation: each synapse alone raises V_dend by 12 mV from rest (−65 mV) → V = −53 mV < −45 mV (no NMDA spike). Both together: V = −65 + 12 + 12 = −41 mV > −45 mV → NMDA spike triggered!

$$\text{Synapse A only: } V_d = -65+12 = -53 < -45 \;\Rightarrow\; \text{no spike}$$ $$\text{Synapse B only: } V_d = -65+12 = -53 < -45 \;\Rightarrow\; \text{no spike}$$ $$\text{Both: } V_d = -65+12+12 = -41 > -45 \;\Rightarrow\; \text{NMDA spike → large somatic EPSP}$$

The AND gate has a 'threshold' (superlinear summation). Somatic EPSP: each synapse alone → 0.3 mV; both together (via NMDA spike) → 4 mV (13× superlinear). This superlinearity was directly measured by Losonczy & Magee (2006) in CA1 pyramidal neurons using 2-photon glutamate uncaging.

▸ References
[PM01]Poirazi & Mel — Impact of active dendrites and structural plasticity on the memory capacity of neural tissue, Neuron 29:779, 2001
[LM06]Losonczy & Magee — Integrative properties of radial oblique dendrites in hippocampal CA1 pyramidal neurons, Neuron 50:291, 2006
[London05]London & Häusser — Dendritic computation, Annu. Rev. Neurosci. 28:503, 2005
[Koc99]Koch — Biophysics of Computation, Oxford, 1999. Ch. 11 (dendrites as neural computers)
§ 04
Frequently Asked Questions
🔬 SimulationWhat does the AND Gate tab show — how do I observe nonlinear summation?▼
The AND Gate tab shows two synapses A and B on the same dendritic branch. First, activate synapse A alone (press 'A only'): the somatic voltage response is small (~0.3 mV). Then activate B alone: similar small response. Then activate both simultaneously ('A+B'): if the combined input exceeds V_NMDA_thresh, a dendritic NMDA spike fires, producing a much larger somatic response (~4 mV, 13× superlinear). The nonlinearity tab sweeps through different input combinations and plots the somatic response as a heat map — superlinear summation appears as a bright region where both inputs are co-active. Compare passive (NMDA blocked) vs active (NMDA enabled) modes.
Key takeaway: Passive dendrites: linear summation (EPSP_A + EPSP_B = EPSP_A+B). Active dendrites (NMDA): superlinear (EPSP_A+B > EPSP_A + EPSP_B when both exceed NMDA threshold). AND gate = superlinear summation from NMDA spike.
🧠 ConceptualWhy are active dendrites computationally equivalent to a 2-layer neural network?▼
Poirazi & Mel (2001) showed that a single L5 pyramidal neuron with ~10 active dendritic branches, each implementing a nonlinear (sigmoidal) integration, is equivalent to a 2-layer feedforward network with 10 hidden units. The first 'layer' is the dendritic branches: each branch performs a nonlinear summation of its synaptic inputs. The second 'layer' is the soma: it linearly sums the dendritic outputs and applies a threshold (spiking). This means a single neuron with 10,000 synapses distributed across active dendrites can classify input patterns with the same power as a small artificial neural network. The classical model (all synapses directly on soma) has much less capacity.
Key takeaway: Single pyramidal neuron with active dendrites ≈ 2-layer ANN. Each dendritic branch = one hidden unit implementing sigmoidal integration. The soma = output layer summing branch activations. This dramatically increases per-neuron computational capacity.
🌍 AppliedHow do dendritic computations relate to deep learning and neuromorphic AI?▼
Active dendrites in biological neurons directly inspired the 'dendritic computing' architecture in modern neuromorphic hardware. Intel's Loihi 2 supports 'graded activation functions' that approximate the sigmoidal dendritic integration. Research groups (Poirazi, Hausser, Bhatt, Richards) have proposed 'dendritic error learning' — using dendritic plateau potentials as local error signals to implement backpropagation in spiking neurons, solving the biological credit assignment problem. Deep learning researchers are now building 'dendritic neural networks' (Guerguiev et al. 2017; Sacramento et al. 2018) where each artificial neuron has a 'basal' compartment (bottom-up input) and 'apical' compartment (top-down feedback), mimicking L5 pyramidal neuron anatomy.
Key takeaway: Active dendrites inspired dendritic computing in neuromorphic hardware. Dendritic error learning proposes a biological implementation of backpropagation using Ca²⁺ plateau potentials as local error signals — solving the credit assignment problem.
💡 Non-ObviousWhy can a single dendritic branch implement XOR, which a linear neuron cannot?▼
A linear (perceptron) neuron cannot solve XOR because XOR is not linearly separable. But a single dendritic branch with nonlinear NMDA summation CAN solve XOR: make the branch subthreshold for each input alone, but also make it subthreshold for BOTH inputs (too much input → shunting inhibition clamps the branch). Only one input at a time → NMDA threshold crossed → NMDA spike. Both inputs → shunting prevents spike. This produces the XOR truth table. Bhatt (2010) and Cazé (2013) showed computationally that active dendritic branches can implement arbitrary Boolean functions, including XOR, by combining NMDA-based super-linearity and inhibitory shunting within the same branch.
Key takeaway: Dendritic NMDA spikes + local inhibitory shunting = XOR gate. Active dendrites break the limitation of linear perceptrons by implementing nonlinear Boolean logic locally in each branch — not just at the output (soma).
📐 ComputationalWhat is the minimum compartmental model that captures dendritic computation?▼
The 2-compartment model (soma + dendritic compartment) captures the essential nonlinearity for NMDA-based dendritic spikes. Minimum parameters: g_ax (axial conductance between soma and dendrite), g_NMDA in the dendrite (voltage-gated), g_AMPA in the dendrite (linear), and standard Hodgkin-Huxley in the soma. The critical parameter: g_ax determines how strongly the dendritic spike influences the soma. At g_ax ≈ 1 nS: weak coupling, dendritic spikes are local; at g_ax ≈ 10 nS: strong coupling, dendritic spikes reliably trigger somatic APs. For detailed compartmental simulations, use NEURON (neuron.yale.edu) with morphologically reconstructed neurons (NeuroMorpho.org provides >100,000 morphologies).
Key takeaway: Minimum model: 2 compartments (soma + dendrite) with NMDA in dendrite and HH in soma, coupled by g_ax. Critical: g_ax determines whether dendritic spikes are local (low g_ax) or reliably trigger somatic APs (high g_ax).
🎓 DeepWhat is the predictive coding model of dendritic computation (Hawkins/Richards)?▼
Jeff Hawkins (Numenta) and Blake Richards (Montreal) proposed that the apical and basal dendrites of L5 pyramidal neurons implement predictive coding: the basal dendrite integrates bottom-up sensory information (feedforward); the apical dendrite integrates top-down predictions (feedback from higher cortical areas). When both coincide (bottom-up matches top-down prediction), the cell is less likely to fire (prediction confirmed — no error signal needed). When they mismatch (bottom-up ≠ top-down), the apical-basal interaction triggers a burst of APs (prediction error signal). This scheme provides a biologically plausible mechanism for backpropagation: apical signals carry the error derivative; basal signals carry the input; their product (implemented by the dendritic coincidence) computes the gradient update.
Key takeaway: Predictive coding: basal dendrite = bottom-up input; apical dendrite = top-down prediction. Their coincidence (or mismatch) implements local gradient computation — a biologically plausible backpropagation mechanism.
§ 04 Best Resources
§ 05
Misconceptions & Common Errors
Sub-block A — Conceptual Misconceptions
❌"Dendritic spines are just passive recipients of synaptic input — their shape doesn't matter."
✅Spine geometry is computationally significant. The thin spine neck (resistance ~100–500 MΩ) electrically isolates the spine head from the dendrite. This means: (1) the spine head can reach high local voltages that would be unattainable if directly on the dendrite (NMDA spike initiation threshold is more easily reached); (2) Ca²⁺ influx during plasticity is confined to the individual spine head, providing synapse-specific rather than branch-wide LTP; (3) spine neck conductance changes (by actin remodelling) provide a 'structural' form of synaptic plasticity independent of AMPA receptor trafficking. Spine morphology dynamically changes during LTP: spine heads enlarge and neck conductance increases, reducing attenuation and increasing synaptic weight.
📖 Bhatt et al. (2009) Annu. Rev. Physiol.; Korkotian & Segal (2001); Koch — Biophysics of Computation, Ch. 11
Sub-block B — Numerical Errors
❌Simulating dendritic computation with a single-compartment soma model — all synapses directly on the soma, no dendritic cable.
✅A single-compartment model removes all dendritic computation: every synapse is equidistant from the soma, all inputs sum linearly, no NMDA spikes are possible, and location-dependent plasticity rules (STDP via back-propagating APs) cannot be implemented. For any research involving dendritic nonlinearity, synaptic location effects, or dendritic Ca²⁺ dynamics, use at least a 2-compartment model (soma + dendrite). For quantitative studies, use NEURON with multi-compartmental morphology. Even pedagogically, a 2-compartment model adds only 1 ODE and 2 parameters but dramatically changes the computational properties.
🔍 Why: Any study of dendritic integration requires at least 2 compartments. Single-compartment models are valid for studying spike timing, rate coding, and network dynamics — but not dendritic computation or location-dependent plasticity.
§ 05 References