eprop_iaf_psc_delta – Current-based leaky integrate-and-fire neuron model with delta-shaped postsynaptic currents for e-prop plasticity

Description

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

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

Note

The neuron dynamics of the eprop_iaf_psc_delta model (excluding e-prop plasticity) are similar to the neuron dynamics of the iaf_psc_delta model, with minor differences, such as the propagator of the post-synaptic current and the voltage reset upon a spike.

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

\[\begin{split}v_j^t &= \alpha v_j^{t-1} + \sum_{i \neq j} W_{ji}^\text{rec} z_i^{t-1} + \sum_i W_{ji}^\text{in} x_i^t \,, \\ \alpha &= e^{ -\frac{ \Delta t }{ \tau_\text{m} } } \,, \\\end{split}\]

where \(W_{ji}^\text{rec}\) and \(W_{ji}^\text{in}\) are the recurrent and input synaptic weight matrices, and \(z_i^{t-1}\) is the recurrent presynaptic state variable, while \(x_i^t\) represents the input at time \(t\).

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

The spike state variable is expressed by a Heaviside function:

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

If the membrane voltage crosses the threshold voltage \(v_\text{th}\), a spike is emitted and the membrane voltage is reset to \(v_\text{reset}\). After the time step of the spike emission, the neuron is not able to spike for an absolute refractory period \(t_\text{ref}\) during which the membrane potential stays clamped to the reset voltage \(v_\text{reset}\), thus

\[v_m = v_\text{reset} \quad \text{for} \quad t_\text{spk} \leq t \leq t_\text{spk} + t_\text{ref} \,.\]

Spikes arriving while the neuron is refractory are discarded by default. However, if refractory_input is set to True they are damped for each time step until the end of the refractory period and then added to the membrane voltage.

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 \(z_i^{t-2}\), the surrogate gradient or pseudo-derivative of the spike state variable with respect to the postsynaptic membrane voltage \(\psi_j^{t-1}\) (the product of which forms the eligibility trace \(e_{ji}^{t-1}\)), and the learning signal \(L_j^t\) emitted by the readout neurons.

Surrogate gradients help overcome the challenge of the spiking function’s non-differentiability, facilitating the use of gradient-based learning techniques such as e-prop. The non-existent derivative of the spiking variable with respect to the membrane voltage, \(\frac{\partial z^t_j}{ \partial v^t_j}\), can be effectively replaced with a variety of surrogate gradient functions, as detailed in various studies (see, e.g., [3]). NEST currently provides four different surrogate gradient functions:

1. A piecewise linear function used among others in [1]:

\[\begin{split}\psi_j^t = \frac{ \gamma }{ v_\text{th} } \text{max} \left( 0, 1-\beta \left| \frac{ v_j^t - v_\text{th} }{ v_\text{th} }\right| \right) \,. \\\end{split}\]

2. An exponential function used in [4]:

\[\begin{split}\psi_j^t = \gamma \exp \left( -\beta \left| v_j^t - v_\text{th} \right| \right) \,. \\\end{split}\]

3. The derivative of a fast sigmoid function used in [5]:

\[\begin{split}\psi_j^t = \gamma \left( 1 + \beta \left| v_j^t - v_\text{th} \right| \right)^2 \,. \\\end{split}\]

4. An arctan function used in [6]:

\[\begin{split}\psi_j^t = \frac{\gamma}{\pi} \frac{1}{ 1 + \left( \beta \pi \left( v_j^t - v_\text{th} \right) \right)^2 } \,. \\\end{split}\]

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:

\[\begin{split}\frac{ \text{d} E^t }{ \text{d} W_{ji} } &= L_j^t \bar{e}_{ji}^{t-1} \,, \\ e_{ji}^{t-1} &= \psi_j^{t-1} \bar{z}_i^{t-2} \,, \\\end{split}\]

The eligibility trace and the presynaptic spike trains are low-pass filtered with the following exponential kernels:

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

Furthermore, a firing rate regularization mechanism keeps the exponential moving average of the postsynaptic neuron’s firing rate \(f_j^{\text{ema},t}\) close to a target firing rate \(f^\text{target}\). The gradient \(g_\text{reg}^t\) of the regularization loss \(E_\text{reg}^t\) with respect to the synaptic weight \(W_{ji}\) is given by:

\[\begin{split}\frac{ \text{d} E_\text{reg}^t }{ \text{d} W_{ji}} &\approx c_\text{reg} \left( f^{\text{ema},t}_j - f^\text{target} \right) \bar{e}_{ji}^t \,, \\ f^{\text{ema},t}_j &= \mathcal{F}_{\kappa_\text{reg}} \left( \frac{z_j^t}{\Delta t} \right) = \kappa_\text{reg} f^{\text{ema},t-1}_j + \left( 1 - \kappa_\text{reg} \right) \frac{z_j^t}{\Delta t} \,, \\\end{split}\]

where \(c_\text{reg}\) is a constant scaling factor.

The overall gradient is given by the addition of the two gradients.

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:

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

For more information on the implementation details of the neuron model, see [7] and [8].

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

• eprop_iaf_psc_delta_adapt

• eprop_readout

• 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	\(E_\text{L}\)	-70.0	Leak / resting membrane potential
I_e	pA	\(I_\text{e}\)	0.0	Constant external input current
t_ref	ms	\(t_\text{ref}\)	2.0	Duration of the refractory period
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
V_th	mV	\(v_\text{th}\)	-55.0	Spike threshold voltage
V_reset	mV	\(v_\text{reset}\)	-70.0	Reset voltage
refractory_input	Boolean		False	If True, spikes arriving during the refractory period are damped until it ends and then added to the membrane voltage

E-prop parameters
Parameter	Unit	Math equivalent	Default	Description
c_reg		\(c_\text{reg}\)	0.0	Coefficient of firing rate regularization
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
f_target	Hz	\(f^\text{target}\)	10.0	Target firing rate of rate regularization
kappa		\(\kappa\)	0.97	Low-pass filter of the eligibility trace
kappa_reg		\(\kappa_\text{reg}\)	0.97	Low-pass filter of the firing rate for regularization
beta		\(\beta\)	1.0	Width scaling of surrogate gradient / pseudo-derivative of membrane voltage
gamma		\(\gamma\)	0.3	Height scaling of surrogate gradient / pseudo-derivative of membrane voltage
surrogate_gradient_function		\(\psi\)	“piecewise_linear”	Surrogate gradient / pseudo-derivative function [“piecewise_linear”, “exponential”, “fast_sigmoid_derivative” , “arctan”]

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	\(v_j\)	-70.0	Membrane voltage

E-prop state variables and recordables
State variable	Unit	Math equivalent	Initial value	Description
learning_signal	pA	\(L_j\)	0.0	Learning signal
surrogate_gradient		\(\psi_j\)	0.0	Surrogate gradient / pseudo-derivative of membrane voltage

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] (1,2,3)

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)

[3]

Neftci EO, Mostafa H, Zenke F (2019). Surrogate Gradient Learning in Spiking Neural Networks. IEEE Signal Processing Magazine, 36(6), 51-63. https://doi.org/10.1109/MSP.2019.2931595

[4]

Shrestha SB, Orchard G (2018). SLAYER: Spike Layer Error Reassignment in Time. Advances in Neural Information Processing Systems, 31:1412-1421. https://proceedings.neurips.cc/paper_files/paper/2018/hash/82.. rubric:: References

[5]

Zenke F, Ganguli S (2018). SuperSpike: Supervised Learning in Multilayer Spiking Neural Networks. Neural Computation, 30:1514–1541. https://doi.org/10.1162/neco_a_01086

[6]

Fang W, Yu Z, Chen Y, Huang T, Masquelier T, Tian Y (2021). Deep residual learning in spiking neural networks. Advances in Neural Information Processing Systems, 34:21056–21069. https://proceedings.neurips.cc/paper/2021/hash/afe434653a898da20044041262b3ac74-Abstract.html

[7]

Rotter S, Diesmann M (1999). Exact simulation of time-invariant linear systems with applications to neuronal modeling. Biological Cybernetics 81:381-402. https://doi.org/10.1007/s004220050570

[8]

Diesmann M, Gewaltig MO, Rotter S, Aertsen A (2001). State space analysis of synchronous spiking in cortical neural networks. Neurocomputing 38-40:565-571. https://doi.org/10.1016/S0925-2312(01)00409-X

Sends

SpikeEvent

Receives

SpikeEvent, CurrentEvent, LearningSignalConnectionEvent, DataLoggingRequest

See also

Examples using this model

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

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