eprop_iaf_adapt_bsshslm_2020 – Current-based leaky integrate-and-fire neuron model with delta-shaped or exponentially filtered postsynaptic currents and threshold adaptation for e-prop plasticity

Description

eprop_iaf_adapt_bsshslm_2020 is an implementation of a leaky integrate-and-fire neuron model with delta-shaped postsynaptic currents and threshold adaptation used for eligibility propagation (e-prop) plasticity.

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

The suffix _bsshslm_2020 follows the NEST convention to indicate in the model name the paper that introduced it by the first letter of the authors’ last names and the publication year.

Note

The neuron dynamics of the eprop_iaf_adapt_bsshslm_2020 model (excluding e-prop plasticity and the threshold adaptation) 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:

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

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 threshold adaptation is given by:

\[\begin{split}A_j^t &= v_\text{th} + \beta a_j^t \,, \\ a_j^t &= \rho a_j^{t-1} + z_j^{t-1} \,, \\ \rho &= e^{-\frac{ \Delta t }{ \tau_\text{a} }} \,. \\\end{split}\]

The spike state variable is expressed by a Heaviside function:

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

If the membrane voltage crosses the adaptive threshold voltage \(A_j^t\), a spike is emitted and the membrane voltage is reduced by \(v_\text{th}\) in the next time step. After the time step of the spike emission, the neuron is not able to spike for an absolute refractory period \(t_\text{ref}\).

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 \(g\) of the loss \(E\) with respect to the synaptic weight \(W_{ji}\): \(\frac{ \text{d}E }{ \text{d} W_{ij} }\) which depends on the presynaptic spikes \(z_i^{t-1}\), the surrogate gradient or pseudo-derivative of the spike state variable with respect to the postsynaptic membrane voltage \(\psi_j^t\) (the product of which forms the eligibility trace \(e_{ji}^t\)), and the learning signal \(L_j^t\) emitted by the readout neurons.

\[\begin{split}\frac{ \text{d} E }{ \text{d} W_{ji} } &= \sum_t L_j^t \bar{e}_{ji}^t \,, \\ e_{ji}^t &= \psi_j^t \left( \bar{z}_i^{t-1} - \beta \epsilon_{ji,a}^{t-1} \right) \,, \\ \epsilon^{t-1}_{ji,\text{a}} &= \psi_j^{t-1} \bar{z}_i^{t-2} + \left( \rho - \psi_j^{t-1} \beta \right) \epsilon^{t-2}_{ji,a} \,. \\\end{split}\]

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]:

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

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

\[\begin{split}\bar{e}_{ji}^t &= \mathcal{F}_\kappa \left( e_{ji}^t \right) \,, \\ \kappa &= e^{ -\frac{\Delta t }{ \tau_\text{m,out} }} \,, \\ \bar{z}_i^t &= \mathcal{F}_\alpha(z_i^t) \,, \\ \mathcal{F}_\alpha \left( z_i^t \right) &= \alpha \mathcal{F}_\alpha \left( z_i^{t-1} \right) + z_i^t \,, \\ \mathcal{F}_\alpha \left( z_i^0 \right) &= z_i^0 \,, \\\end{split}\]

where \(\tau_\text{m,out}\) is the membrane time constant of the readout neuron.

Furthermore, a firing rate regularization mechanism keeps the average firing rate \(f^\text{av}_j\) of the postsynaptic neuron close to a target firing rate \(f^\text{target}\). The gradient \(g_\text{reg}\) of the regularization loss \(E_\text{reg}\) with respect to the synaptic weight \(W_{ji}\) is given by:

\[\begin{split}\frac{ \text{d} E_\text{reg} }{ \text{d} W_{ji} } = c_\text{reg} \sum_t \frac{ 1 }{ T n_\text{trial} } \left( f^\text{target} - f^\text{av}_j \right) e_{ji}^t \,, \\\end{split}\]

where \(c_\text{reg}\) is a constant scaling factor and the average is taken over the time that passed since the previous update, that is, the number of trials \(n_\text{trial}\) times the duration of an update interval \(T\).

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

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

• eprop_iaf_bsshslm_2020

• eprop_readout_bsshslm_2020

• eprop_synapse_bsshslm_2020

• eprop_learning_signal_connection_bsshslm_2020

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
adapt_beta		\(\beta\)	1.0	Prefactor of the threshold adaptation
adapt_tau	ms	\(\tau_\text{a}\)	10.0	Time constant of the threshold adaptation
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
regular_spike_arrival	Boolean		True	If True, the input spikes arrive at the end of the time step, if False at the beginning (determines PSC scale)
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

E-prop parameters
Parameter	Unit	Math equivalent	Default	Description
c_reg		\(c_\text{reg}\)	0.0	Coefficient of firing rate regularization
f_target	Hz	\(f^\text{target}\)	10.0	Target firing rate of rate 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
adaptation		\(a_j\)	0.0	Adaptation variable
V_m	mV	\(v_j\)	-70.0	Membrane voltage
V_th_adapt	mV	\(A_j\)	-55.0	Adapting spike threshold

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

Sends

SpikeEvent

Receives

SpikeEvent, CurrentEvent, LearningSignalConnectionEvent, DataLoggingRequest

See also

Examples using this model

• 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)