eprop_readout – Current-based leaky integrate readout neuron model with delta-shaped postsynaptic currents for e-prop plasticity

Description

eprop_readout is an implementation of an integrate-and-fire neuron model with delta-shaped postsynaptic currents used as readout neuron for eligibility propagation (e-prop) plasticity.

E-prop plasticity was originally introduced and implemented in TensorFlow in [1].

The membrane voltage time course \(v_j^t\) of the neuron \(j\) is given by:

\[\begin{split}v_j^t &= \kappa v_j^{t-1}+ \zeta \sum_{i \neq j} W_{ji}^\text{out} z_i^{t-1} \,, \\ \kappa &= e^{ -\frac{ \Delta t }{ \tau_\text{m} } } \,, \\ \zeta &= \begin{cases} 1 \\ 1 - \kappa \end{cases} \,, \\\end{split}\]

where \(W_{ji}^\text{out}\) is the output synaptic weight matrix and \(z_i^{t-1}\) is the recurrent presynaptic spike state variable.

Descriptions of further parameters and variables can be found in the table below.

The spike state variable of a presynaptic neuron is expressed by a Heaviside function:

\[\begin{split}z_i^t = H \left( v_i^t - v_\text{th} \right) \,. \\\end{split}\]

An additional state variable and the corresponding differential equation represents a piecewise constant external current.

See the documentation on the iaf_psc_delta neuron model for more information on the integration of the subthreshold dynamics.

The change of the synaptic weight is calculated from the gradient \(g^t\) of the loss \(E^t\) with respect to the synaptic weight \(W_{ji}\): \(\frac{ \text{d} E^t }{ \text{d} W_{ij} }\) which depends on the presynaptic spikes \(z_i^{t-1}\) and the learning signal \(L_j^t\) emitted by the readout neurons.

In the interval between two presynaptic spikes, the gradient is calculated at each time step until the cutoff time point. This computation occurs over the time range:

\(t \in \left[ t_\text{spk,prev}, \min \left( t_\text{spk,prev} + \Delta t_\text{c}, t_\text{spk,curr} \right) \right]\).

Here, \(t_\text{spk,prev}\) represents the time of the previous spike that passed the synapse, while \(t_\text{spk,curr}\) is the time of the current spike, which triggers the application of the learning rule and the subsequent synaptic weight update. The cutoff \(\Delta t_\text{c}\) defines the maximum allowable interval for integration between spikes. The expression for the gradient is given by:

\[\begin{split}\frac{ \text{d} E^t }{ \text{d} W_{ji} } = L_j^t \bar{z}_i^{t-1} \,. \\\end{split}\]

The presynaptic spike trains are low-pass filtered with the following exponential kernel:

\[\begin{split}\bar{z}_i^t = \mathcal{F}_\kappa \left( z_{i}^t \right) = \kappa \bar{z}_i^{t-1} + \zeta z_i^t \,. \\\end{split}\]

Since readout neurons are leaky integrators without a spiking mechanism, the formula for computing the gradient lacks the surrogate gradient / pseudo-derivative and a firing regularization term.

As a last step for every round in the loop over the time steps \(t\), the new weight is retrieved by feeding the current gradient \(g^t\) to the optimizer (see weight_optimizer for more information on the available optimizers):

\[\begin{split}w^t = \text{optimizer} \left( t, g^t, w^{t-1} \right) \,. \\\end{split}\]

After the loop has terminated, the filtered dynamic variables of e-prop are propagated from the end of the cutoff until the next spike:

\[\begin{split}p &= \text{max} \left( 0, t_\text{s}^{t} - \left( t_\text{s}^{t-1} + {\Delta t}_\text{c} \right) \right) \,, \\ \bar{z}_i^{t+p} &= \bar{z}_i^t \alpha^p \,. \\\end{split}\]

The learning signal \(L_j^t\) is given by the non-plastic feedback weight matrix \(B_{jk}\) and the continuous error signal \(e_k^t\) emitted by readout neuron \(k\) and \(e_k^t\) defined via a mean-squared error loss:

\[\begin{split}L_j^t = B_{jk} e_k^t = B_{jk} \left( y_k^t - y_k^{*,t} \right) \,. \\\end{split}\]

where the readout signal \(y_k^t\) corresponds to the membrane voltage of readout neuron \(k\) and \(y_k^{*,t}\) is the real-valued target signal.

Furthermore, the readout and target signal are multiplied by a learning window signal, which has a value of 1.0 within the learning window and 0.0 outside.

For more information on e-prop plasticity, see the documentation on the other e-prop models:

• eprop_iaf

• eprop_iaf_adapt

• eprop_synapse

• eprop_learning_signal_connection

Details on the event-based NEST implementation of e-prop can be found in [2].

Parameters

The following parameters can be set in the status dictionary.

Neuron parameters
Parameter	Unit	Math equivalent	Default	Description
C_m	pF	\(C_\text{m}\)	250.0	Capacitance of the membrane
E_L	mV	\(E_\text{L}\)	0.0	Leak / resting membrane potential
I_e	pA	\(I_\text{e}\)	0.0	Constant external input current
tau_m	ms	\(\tau_\text{m}\)	10.0	Time constant of the membrane
V_min	mV	\(v_\text{min}\)	negative maximum value representable by a double type in C++	Absolute lower bound of the membrane voltage

E-prop parameters
Parameter	Unit	Math equivalent	Default	Description
eprop_isi_trace_cutoff	ms	\({\Delta t}_\text{c}\)	maximum value representable by a long type in C++	Cutoff for integration of e-prop update between two spikes

Recordables

The following state variables evolve during simulation and can be recorded.

Neuron state variables and recordables
State variable	Unit	Math equivalent	Initial value	Description
V_m	mV	\(v_j\)	0.0	Membrane voltage

E-prop state variables and recordables
State variable	Unit	Math equivalent	Initial value	Description
error_signal	mV	\(L_j\)	0.0	Error signal
readout_signal	mV	\(y_j\)	0.0	Readout signal
target_signal	mV	\(y^*_j\)	0.0	Target signal

Usage

This model can only be used in combination with the other e-prop models and the network architecture requires specific wiring, input, and output. The usage is demonstrated in several supervised regression and classification tasks reproducing among others the original proof-of-concept tasks in [1].

References

[1] (1,2)

Bellec G, Scherr F, Subramoney F, Hajek E, Salaj D, Legenstein R, Maass W (2020). A solution to the learning dilemma for recurrent networks of spiking neurons. Nature Communications, 11:3625. https://doi.org/10.1038/s41467-020-17236-y

[2]

Korcsak-Gorzo A, Stapmanns J, Espinoza Valverde JA, Plesser HE, Dahmen D, Bolten M, Van Albada SJ, Diesmann M. Event-based implementation of eligibility propagation (in preparation)

Sends

LearningSignalConnectionEvent, DelayedRateConnectionEvent

Receives

SpikeEvent, CurrentEvent, DelayedRateConnectionEvent, DataLoggingRequest

See also

Examples using this model

• Tutorial on learning N-MNIST classification with e-prop

• Tutorial on learning to accumulate evidence with e-prop after Bellec et al. (2020)

• Tutorial on learning to generate a lemniscate with e-prop after Bellec et al. (2020)

• Tutorial on learning to generate handwritten text with e-prop after Bellec et al. (2020)

• Tutorial on learning to generate sine waves with e-prop

• Tutorial on learning to generate sine waves with e-prop after Bellec et al. (2020)