Two population STDP network model¶

Network model¶

In this example, we employ a simple network model describing the dynamics of a local cortical circuit at the spatial scale of ~1mm within a single cortical layer. It is derived from the model proposed in [1], but accounts for the synaptic weight dynamics for connections between excitatory neurons. The weight dynamics are described by the spike-timing-dependent plasticity (STDP) model derived in [2]. This network model is used as the basis for scaling experiments comparing the model ignore-and-fire with the iaf_psc_alpha. You can find

Summary of network model¶

Populations	excitatory population \(\mathcal{E}\), inhibitory population \(\mathcal{I}\), external Poissonian spike sources \(\mathcal{X}\)
Connectivity	sparse random connectivity respecting Dale’s principle
Neurons	leaky integrate-and-fire (LIF)
Synapses	linear input integration with alpha-function-shaped postsynaptic currents (PSCs), spike-timing dependent plasticity (STDP) for connections between excitatory neurons
Input	stationary, uncorrelated Poissonian spike trains
	Figure 48 Network sketch (see Fig. 8 in Senk et al. [3]).¶

Detailed desciption of network model¶

Populations¶

Name	Elements	Size
\(\mathcal{E}\)	LIF neurons	\(N_\text{E}=\beta{}N\)
\(\mathcal{I}\)	LIF neurons	\(N_\text{I}=N-N_\text{E}\)
\(\mathcal{X}\)	realizations of a Poisson point process	\(N\)

Connectivity¶

Source	Target	Pattern
\(\mathcal{E}\)	\(\mathcal{E}\)	random, independent; homogeneous in-degree \(K_{\text{E},i}=K_\text{E}\) (\(\forall{}i\in\mathcal{E}\)) plastic synaptic weights \(J_{ij}(t)\) (\(\forall{}i\in\mathcal{E},j\in\mathcal{E}\)) homogeneous spike-transmission delays \(d_{ij}=d\) (\(\forall{}i\in\mathcal{E},j\in\mathcal{E}\))
\(\mathcal{E}\)	\(\mathcal{I}\)	random, independent; homogeneous in-degree \(K_{\text{E},i}=K_\text{E}\) (\(\forall{}i\in\mathcal{I}\)) fixed synaptic weights \(J_{ij}\in\{0,J\}\) (\(\forall{}i\in\mathcal{I},j\in\mathcal{E}\)) homogeneous spike-transmission delays \(d_{ij}=d\) (\(\forall{}i\in\mathcal{I},j\in\mathcal{E}\))
\(\mathcal{I}\)	\(\mathcal \ {E}\cup\mathcal{I}\)	random, independent; homogeneous in-degree \(K_{\text{I},i}=K_\text{I}\) (\(forall{}i\in\mathcal{E}\cup\mathcal{I}\))jinmathcal{I}`) fixed synaptic weights \(J_{ij}\in\{-gJ,0\}\) (\(\forall{}i\in\mathcal{E}\cup\mathcal{I}, \ j\in\mathcal{I}\)) homogeneous spike-transmission delays \(d_{ij}=d\) (\(\forall{}i\in\mathcal{E}\cup\mathcal{I}, \ j\in\mathcal{I}\))
\(\mathcal{X}\)	\(\mathcal \ {E}\cup\mathcal{I}\)	one-to-one fixed synaptic weights \(J_{ij}=J\) (\(\forall{}i\in\mathcal{E}\cup\mathcal{I}, \ j\in\mathcal{X}\)) homogeneous spike-transmission delays \(d_{ij}=d\) (\(\forall{}i\in\mathcal{E}\cup\mathcal{I}, \ j\in\mathcal{X}\))

Neuron¶

Leaky integrate-and-fire (iaf) dynamics

Dynamics of membrane potential \(V_{i}(t)\) and spiking activity \(s_i(t)\) of neuro \(i\in\left\{1,\ldots,N\right\}\):

emission of \(k\)th (\(k=1,2,\ldots\)) spike of neuron \(i\) at time \(t_{i}^{k}\) if

\[V_{i}\left(t_{i}^{k}\right)\geq\theta\]

with spike threshold \(\theta\)
reset and refractoriness:

\[\forall{}k,\ \forall t \in \left(t_{k}^{i},\,t_{k}^{i}+\tau_\text{ref}\right]:\quad V_{i}(t)=V_\text{reset}\]

with refractory period \(\tau_\text{ref}\) and reset potential \(V_\text{reset}\)
spike train \(\displaystyle s_i(t)=\sum_k \delta(t-t_i^k)\)
subthreshold dynamics of membrane potential \(V_{i}(t)\):

\[\begin{split}\begin{aligned} &\forall{}k,\ \forall t \notin \left[t_{i}^{k},\,t_{i}^{k}+\tau_\text{ref}\right):\\ &\qquad\tau_\text{m}\frac{\text{d}{}V_i(t)}{\text{d}{}t} = \Bigl[E_\text{L}-V_i(t)\Bigr]+R_\text{m}I_i(t) \end{aligned}\end{split}\]

with membrane time constant \(\tau_\text{m}\), membrane resistance \(R_\text{m}\), resting potential \(E_\text{L}\), and total synaptic input current \(I_i(t)\)

Synapse: transmission¶

Current-based synapses with alpha-function shaped postsynaptic currents (PSCs)

Total synaptic input current of neuron \(i\)

\[I_i(t)=I_{\text{E},i}(t)+I_{\text{I},i}(t)+I_{\text{X},i}(t)\]

excitatory, inhibitory and external synaptic input currents

\[\begin{split}%I_{P,i}(t)=\sum_{j\in\mathcal{P}}(\text{PSC}_{ij}*s_j)(t) %\quad\text{for}\quad %(P,\mathcal{P})\in\{(\exc,\Epop),(\inh,\Ipop),(\ext,\Xpop)\} %, \begin{aligned} I_{\text{E},i}(t)&=\sum_{j\in\mathcal{E}}\bigl(\text{PSC}_{ij}*s_j\bigr)(t-d_{ij})\\ I_{\text{I},i}(t)&=\sum_{j\in\mathcal{I}}\bigl(\text{PSC}_{ij}*s_j\bigr)(t-d_{ij})\\ I_{\text{X},i}(t)&=\sum_{j\in\mathcal{X}}\bigl(\text{PSC}_{ij}*s_j\bigr)(t-d_{ij}) \end{aligned}\end{split}\]

with spike trains \(s_j(t)\) of local (\(j\in\mathcal{E}\cup\mathcal{I}\)) and external sources (\(j\in\mathcal{X}\)), spike transmission delays \(d_{ij}\), and convolution operator “\(*\)”: \(\displaystyle\bigl(f*g\bigr)(t)=\int_{-\infty}^\infty\text{d}s\,f(s)g(t-s)\))
alpha-function shaped postsynaptic currents

\[\text{PSC}_{ij}(t)=\hat{I}_{ij}e\tau_\text{s}^{-1}te^{-t/\tau_\text{s}}\Theta(t)\]

with synaptic time constant \(\tau_\text{s}\) and Heaviside function \(\Theta(\cdot)\)
postsynaptic potential triggered by a single presynaptic spike

\[\text{PSP}_{ij}(t)= \hat{I}_{ij}\frac{e}{\tau_\text{s}C_\text{m}} \left(\frac{1}{\tau_\text{m}}-\frac{1}{\tau_\text{s}}\right)^{-2} \left(\left(\frac{1}{\tau_\text{m}}-\frac{1}{\tau_\text{s}}\right) t e^{-t/\tau_\text{s}} - e^{-t/\tau_\text{s}} + e^{-t/\tau_\text{m}} \right) \Theta(t)\]
PSC amplitude (synaptic weight)

\[\hat{I}_{ij}=\text{max}_t\bigl(\text{PSC}_{ij}(t)\bigr) =\frac{J_{ij}}{J_\text{unit}(\tau_\text{m},\tau_\text{s},C_\text{m})}\]

parameterized by PSP amplitude \(J_{ij}=\text{max}_t\bigl(\text{PSP}_{ij}(t)\bigr)\)

with unit PSP amplitude (PSP amplitude for \(\hat{I}_{ij}=1\)):

\[J_\text{unit}(\tau_\text{m},\tau_\text{s},C_\text{m}) = \frac{e}{C_\text{m}\left(1-\frac{\tau_\text{s}}{\tau_\text{m}}\right)}\left( \frac{e^{-t_\text{max}/\tau_\text{m}} - e^{-t_\text{max}/\tau_\text{s}}}{\frac{1}{\tau_\text{s}} - \frac{1}{\tau_\text{m}}} - t_\text{max}e^{-t_\text{max}/\tau_\text{s}} \right),\]

time to PSP maximum

\[t_\text{max} = \frac{1}{\frac{1}{\tau_\text{s}} - \frac{1}{\tau_\text{m}}}\left(-W_{-1}\left(\frac{-\tau_\text{s}e^{-\frac{\tau_\text{s}}{\tau_\text{m}}}}{\tau_\text{m}}\right) - \frac{\tau_\text{s}}{\tau_\text{m}}\right),\]

and Lambert-W function \(\displaystyle W_{-1}(x)\) for \(\displaystyle x \ge -1/e\)

Synapse: plasticity¶

Spike-timing dependent plasticity (STDP) with power-law weight dependence and all-to-all spike pairing scheme. See Morrison et al. [2] for connections between excitatory neurons.

Dynamics of synaptic weights \(J_{ij}(t)\) \(\forall{}i\in\mathcal{E}, j\in\mathcal{E}\):

\[\begin{split}\begin{aligned} &\forall J_{ij}\ge{}0: \\[1ex] &\quad \frac{\text{d}}{}J_{ij}{\text{d}{}t}= \lambda^+f^+(J_{ij})\sum_k x^+_j(t)\delta\Bigl(t-[t_i^k+d_{ij}]\Bigr) + \lambda^-f^-(J_{ij})\sum_l x^-_i(t)\delta\Big(t-[t_j^l-d_{ij}]\Bigr)\\[1ex] &\forall{}\{t|J_{ij}(t)<0\}: \quad J_{ij}(t)=0 \quad \text{(clipping)} \end{aligned}\end{split}\]

with

pre- and postsynaptic spike times \(\{t_j^l|l=1,2,\ldots\}\) and \(\{t_i^k|k=1,2,\ldots\}\),

magnitude \(\lambda^+=\lambda\) of weight update for causal firing (postsynaptic spike following presynaptic spikes: \(t_i^k>t_j^l\)),

magnitude \(\lambda^-=-\alpha\lambda\) of weight update for acausal firing (presynaptic spike following postsynaptic spikes: \(t_i^k<t_j^l\)),

power-law weight dependence \(f^+(J_{ij})=J_0(J_{ij}/J_0)^{\mu^+}\) of weight update for causal firing with exponent \(\mu^+\) and reference weight \(J_0\),

linear weight dependence \(f^-(J_{ij})=J_{ij}\) of weight update for acausal firing,

(dendritic) delay \(d_{ij}\),

spike trace \(x^+_j(t)\) of presynaptic neuron \(j\), evolving according to

\[\frac{\text{d}{}x^+_j}{\text{d}{}t}=-\frac{x^+_j(t)}{\tau^+}+\sum_l\delta(t-t_j^l)\]

with presynaptic spike times \(\{t_j^l|l=1,2,\ldots\}\) and time constant \(\tau^+\),

spike trace \(x^-_i(t)\) of postsynaptic neuron \(i\), evolving according to

\[\frac{\text{d}{}x^-_i}{\text{d}{}t}=-\frac{x^-_i(t)}{\tau^-}+\sum_k\delta(t-t_i^k)\]

with postsynaptic spike times \(\{t_i^k|k=1,2,\ldots\}\) and time constant \(\tau^-\)

Note

The above weight update accounts for all pairs of pre- and postsynaptic spikes (all-to-all spike pairing scheme). The spike histories and the dependence of the weight update on the time lag of pre- and postsynaptic firing are fully captured by the spike traces \(x^+_j(t)\) and \(x^-_i(t)\).

Stimulus¶

Type	Description
stationary, uncorrelated Poisson spike trains	\(N=\|\mathcal{X}\|\) independent realizations \(s_i(t)\) (\(i\in\mathcal{X}\)) of a Poisson point process with constant rate \(\nu_\text{X}(t)=\eta\nu_\theta\), where \[ \begin{align}\begin{aligned}\label{eq:rheobase_rate_LIF_alpha}\\ \nu_\theta=\frac{\theta-E _\text{L}}{R_\text{m}{} \hat{I}_X{}e\tau_\text{s}}\end{aligned}\end{align} \] denotes the rheobase rate, and \(\eta\) and \(\hat{I}_X=J/J_\text{unit}\) the relative rate and the synaptic weight (PSC amplitude) of external sources

Type

Description

stationary, uncorrelated Poisson spike trains

\(N=|\mathcal{X}|\) independent realizations \(s_i(t)\) (\(i\in\mathcal{X}\)) of a Poisson point process with constant rate \(\nu_\text{X}(t)=\eta\nu_\theta\), where

\[ \begin{align}\begin{aligned}\label{eq:rheobase_rate_LIF_alpha}\\ \nu_\theta=\frac{\theta-E _\text{L}}{R_\text{m}{} \hat{I}_X{}e\tau_\text{s}}\end{aligned}\end{align} \]

denotes the rheobase rate, and \(\eta\) and \(\hat{I}_X=J/J_\text{unit}\) the relative rate and the synaptic weight (PSC amplitude) of external sources

Initial conditions¶

Type	Description
random initial membrane potentials, homogeneous initial synaptic weights and spike traces	membrane potentials: \(V_i(t=0)\sim \ \mathcal{U}(V_{0,\text{min}},V_{0,\text{max}})\) randomly and independently drawn from a uniform distribution between \(V_{0,\text{min}}\) and \(V_{0,\text{max}}\) (\(\forall{}i\)) synaptic weights: \(\hat{I}_{ij}(t=0)=J/J_\text{unit}\) (\(\forall{}i\in\mathcal{E}, \ j\in\mathcal{E}\)) spike traces: \(x_{+,i}(t=0)=x_{-,i}(t=0)=0\) (\(\forall{}i\in\mathcal{E}\))

Type

Description

random initial membrane potentials, homogeneous initial synaptic weights and spike traces

membrane potentials: \(V_i(t=0)\sim \ \mathcal{U}(V_{0,\text{min}},V_{0,\text{max}})\) randomly and independently drawn from a uniform distribution between \(V_{0,\text{min}}\) and \(V_{0,\text{max}}\) (\(\forall{}i\))
synaptic weights: \(\hat{I}_{ij}(t=0)=J/J_\text{unit}\)

(\(\forall{}i\in\mathcal{E}, \ j\in\mathcal{E}\))
spike traces: \(x_{+,i}(t=0)=x_{-,i}(t=0)=0\) (\(\forall{}i\in\mathcal{E}\))

Model parameters¶