Multicompartment neurons

Tripod (AdEx soma, two passive dendrites d1, d2) and BallAndStick (one dendrite d).

Group	Population field	Type	Default
morphology	`param`	`DendNeuronParameter`	`TripodParameter()` / `BallAndStickParameter()`
soma	`adex`	`AdExParameter`	`AdExParameter()`
spike shape, refractoriness	`spike`	`PostSpike`	`PostSpike()`
synapses	`soma_syn`, `dend_syn`	`AbstractSynapseParameter`	`TripodSomaSynapse`, `TripodDendSynapse`

A connection selects the compartment (:s, :d1, :d2; :s, :d) with its fourth argument:

using SpikingNeuralNetworks
SNN.@load_units
T = SNN.Tripod(N = 10)
E = SNN.Poisson(N = 50, param = SNN.PoissonParameter(20Hz))
I = SNN.Poisson(N = 50, param = SNN.PoissonParameter(20Hz))
E_d1 = SNN.SpikingSynapse(E, T, :glu, :d1; conn = (p = 0.2, μ = 2.0))   # first dendrite
E_d2 = SNN.SpikingSynapse(E, T, :glu, :d2; conn = (p = 0.2, μ = 2.0))   # second dendrite
I_s  = SNN.SpikingSynapse(I, T, :gaba, :s; conn = (p = 0.2, μ = 2.0))   # soma
model = SNN.compose(; E, I, T, E_d1, E_d2, I_s)
SNN.monitor!(T, [:v_s, :v_d1, :v_d2, :fire])
SNN.sim!(; model, duration = 500ms)

Equations

For dendrite $k$ ($k = 1, 2$ for Tripod, $k = 1$ for BallAndStick), with somatic parameters from adex and spike parameters from spike:

\[\begin{aligned} C \frac{dV_s}{dt} &= g_L (E_L - V_s) + g_L \Delta_T\, e^{(V_s - \theta)/\Delta_T} - w_s - I_{syn,s} - \sum_k g_{ax,k}\,(V_s - V_{d,k}) + I \\ C_{d,k} \frac{dV_{d,k}}{dt} &= g_{m,k} (E_{L,k} - V_{d,k}) - I_{syn,d,k} + g_{ax,k}\,(V_s - V_{d,k}) + I_d \\ \tau_w \frac{dw_s}{dt} &= a (V_s - E_L) - w_s \\ \tau_A \frac{d\theta}{dt} &= V_t - \theta \end{aligned}\]

$I_{syn}$ are the currents of the synapse models (clamped to $\pm 1500$ pA in Tripod, $\pm 1000$ pA in BallAndStick); $C_{d,k}$, $g_{m,k}$, $g_{ax,k}$ come from create_dendrite.

A spike is emitted when $V_s$ reaches Vspike (default $-10$ mV): $V_s$ is held at spike.AP_membrane for spike.up, then at adex.Vr for spike.τabs; $w_s \mathrel{+}= b$, $\theta \mathrel{+}= A_t$. Integration: Heun method.

Parameters

Morphology: `DendNeuronParameter`

Field	Default	Units	Meaning
`ds`	`[(200um, 400um), (200um, 400um)]`	cm	one entry per dendrite: a length or a `(min, max)` range sampled per neuron in 1 μm steps
`physiology`	`human_dend`		cable properties (`Physiology`)
`geometry`	`[(:s=>:d1), (:s=>:d2)]`		compartment graph, stored for bookkeeping
`type`	`TripodNeuron()` if `length(ds) == 2`, `BallAndStickNeuron()` if `length(ds) == 1`		other lengths raise an error

Shortcuts: TripodParameter, BallAndStickParameter. Predefined length ranges: proximal_distal, proximal_proximal, proximal, all_lengths.

Cable properties: `Physiology`

Field	Meaning	`human_dend`	`mouse_dend`
`Ri`	axial resistivity	200 Ω cm	200 Ω cm
`Rd`	specific membrane resistance	38907 Ω cm²	1700 Ω cm²
`Cd`	specific membrane capacitance	0.5 μF/cm²	1 μF/cm²

For a cylinder of length $l$ and diameter $d$:

\[G_{ax} = \frac{\pi d^2}{4 R_i l} \;(\texttt{G\_axial}), \qquad G_m = \frac{\pi d l}{R_d} \;(\texttt{G\_mem}), \qquad C = C_d\, \pi d l \;(\texttt{C\_mem}).\]

using SpikingNeuralNetworks
SNN.@load_units
SNN.G_axial(Ri = 200Ω*cm, d = 4um, l = 200um)       # ≈ 31.4 nS
d = SNN.create_dendrite(10, (150um, 300um); physiology = SNN.mouse_dend)
d.C[1], d.gm[1], d.gax[1]

Spike: `PostSpike` fields used

Field	Default	Units	Meaning
`AP_membrane`	`10mV`	mV	somatic potential during the spike
`up`	`1ms`	ms	duration of the clamped spike (back-propagation period)
`τabs`	`1ms`	ms	absolute refractory period after `up`
`At`	`0mV`	mV	threshold increment $A_t$ per spike
`τA`	`10ms`	ms	relaxation time constant of $\theta$

Synapses

Defaults: soma_syn = TripodSomaSynapse (AMPA + GABAa), dend_syn = TripodDendSynapse (AMPA + NMDA + GABAa + GABAb); see Synapse and receptor models.

Tripod

Reference: Quaresima et al. (2023), J. Physiol.

using SpikingNeuralNetworks
SNN.@load_units
T = SNN.Tripod(N = 10, param = SNN.TripodParameter(ds = SNN.proximal_distal),
               adex = SNN.AdExParameter(b = 0pA, a = 0nS))
T.I .= 300pA
SNN.monitor!(T, [:v_s, :fire])
SNN.sim!([T]; duration = 200ms)

SNNModels.Tripod — Type

Tripod(; N = 100, param = TripodParameter(), adex = AdExParameter(),
         soma_syn = TripodSomaSynapse, dend_syn = TripodDendSynapse, spike = PostSpike(), ...)

Population of three-compartment neurons: an AdEx-like soma coupled to two passive dendrites (d1, d2) through axial conductances. The dendritic geometry and cable properties are given by param (a DendNeuronParameter), the somatic parameters by adex, the somatic and dendritic synapses by soma_syn and dend_syn (any AbstractSynapseParameter with a five-argument synaptic_current!, i.e. not DeltaSynapse).

Connections select the compartment with the fourth argument of SpikingSynapse: SpikingSynapse(pre, T, :glu, :d1; conn), with targets :s, :d1, :d2.

Equations

Soma (AdEx-like, parameters from adex, an AdExParameter), for each dendrite $k$:

$I_{syn,s}$ and $I_{syn,d,k}$ are computed by synaptic_current! of soma_syn and dend_syn with the compartment potentials of the Heun stage being evaluated, and are clamped to $\pm 1500$ pA (Tripod) or $\pm 1000$ pA (BallAndStick). $C_{d,k}$, $g_{m,k}$, $g_{ax,k}$ and the dendritic leak reversal $E_{L,k}$ come from the Dendrite structs (see create_dendrite); by default $E_{L,k}$ is the somatic $E_L$ (adex.El). The exponential term is multiplied by $g_L$, as in the AdEx model (Brette and Gerstner 2005) and in the published Tripod model (Quaresima et al. 2023, Eq. 1).

Spike: when the predicted somatic potential $V_s + dt\,\dot V_s$ reaches Vspike (population field, default $-10$ mV; adex.Vt is only the resting value of $\theta$), the neuron fires: $V_s \leftarrow$ AP_membrane, $w_s \leftarrow w_s + b$, $\theta \leftarrow \theta + A_t$, and the refractory counter is set to n_up + n_abs steps, with n_up = max(1, round(up/dt)) and n_abs = max(1, round(τabs/dt)). The soma is clamped at AP_membrane until the counter reaches n_abs (back-propagation period; the spike step itself counts as one), then at adex.Vr for n_abs steps; in both periods the dendrites only relax towards the soma through the axial term (forward Euler) and w_s is not integrated. $\theta$ relaxes to adex.Vt at every step (forward Euler). The published Tripod code (TripodNeuron.jl) detects spikes at $V_s \ge V_T$; Vspike = adex.Vt reproduces that rule.

Integration

Heun (explicit trapezoidal) method on $x = (V_s, V_{d,k}, w_s)$. Synaptic conductances are advanced first, once per step, by update_synapses!. Then $k_1 = f(x_n)$ is evaluated at the current state (stored in Δv_temp) and $k_2 = f(x_n + dt\,k_1)$ at the Euler-predicted state (stored in Δv); every term of $f$, including the synaptic currents, the axial currents and the adaptation current in the somatic equation, is evaluated at the same stage state. The state is advanced by $x_{n+1} = x_n + \frac{dt}{2}(k_1 + k_2)$. The scheme is second-order accurate between spikes.

Fields

Population info

id::String = randstring(12), name::String = "Tripod", N::IT = 100, records::Dict.

Parameters

param::DendNeuronParameter = TripodParameter(): morphology (ds, physiology).
adex::SOMAT = AdExParameter(): somatic parameters; uses C, gl, El, ΔT, Vt (resting value of the threshold $\theta$), Vr, a, b, τw.
soma_syn::SYNS = TripodSomaSynapse, dend_syn::SYND = TripodDendSynapse: synapse models.
spike::PST = PostSpike(): AP_membrane, up, τabs, At, τA.
d1::VDT, d2::VDT: Dendrites built with create_dendrite(N, param.ds[k]; El = adex.El, FT = eltype(VFT), physiology = param.physiology).
Vspike::Float32 = -10mV: spike detection threshold on the predicted somatic potential (Float32 in both precisions).

State variables

v_s, v_d1, v_d2::VFT: compartment potentials (mV), initialised uniformly in [Vr, Vt]. VFT = Vector{Float64} gives a Float64 population (see float_type): adex, spike, the synapse models and the dendrites follow, with the decimal default values; param (morphology) and Vspike stay Float32.
w_s::VFT: somatic adaptation current (pA), zeros.
I::VFT: external current into the soma (pA); I_d::VFT: external current injected in each of the two dendrites (pA).
fire::VBT: spike flags; tabs::VFT: refractory counters (steps); θ::VFT: dynamic threshold of the exponential term (mV), initialised to Vt.
spikes::SpikeIndex: neurons that fired in the step (see spike_index).

Synapses

synvars_s, synvars_d1, synvars_d2: synaptic state variables (synaptic_variables).
receptors_s, receptors_d1, receptors_d2::NamedTuple: input buffers (synaptic_receptors).

Work arrays

Δv, Δv_temp::MFT (N x 4): derivatives of (v_s, v_d1, v_d2, w_s) at the two Heun stages.
is::MFT (N x 3): synaptic currents of soma, d1, d2; ic::VFT (length 2): unused, kept for compatibility; v_pred::MFT (N x 3): stage potentials of soma, d1, d2 (mV).

References

Quaresima A. et al. (2023), "The Tripod neuron: a minimal structural reduction of the dendritic tree", J. Physiol. Receptor parameters: see TripodSomaSynapse and TripodDendSynapse.

Example

using SpikingNeuralNetworks
SNN.@load_units
T = SNN.Tripod(N = 10)
E = SNN.Poisson(N = 50, param = SNN.PoissonParameter(20Hz))
syn = SNN.SpikingSynapse(E, T, :glu, :d1; conn = (p = 0.2, μ = 2.0))
model = SNN.compose(; E, T, syn)
SNN.monitor!(T, [:v_s, :v_d1, :fire])
SNN.sim!(; model, duration = 200ms)