Training in-the-loop

To train the network without on-chip plasticity, we used the idea of in-the-loop training (Schmuker et al. [2014], Esser et al. [2016], Schmitt et al. [2017]; sec- tion 2.3.3). As discussed before, the analog parameters are only written once at the beginning of the experiment. During the training procedure, we only change digital parameters, namely the weights between the sampling neurons and the bias connections from the bias neurons to the samplers. This resulted in a larger experiment-repetition frequency and therefore faster training speed in terms of wall-clock time spent compared to the case if we changed the analog parameters.

For the updates, we configure the network, we execute the experiment on BSS-1 and read out the spike-trains of the sampling neurons. We turn the spike-trains into states according to LIF Sampling theory and we calculate the parameter updates with the CD rule. During training, the parameter values are stored in double precision float numbers; and for experiment execution we discretize them deterministically to the nearest available 4-bit value. In machine learning, this is called the method of “shadow weights” [Courbariaux et al., 2015]. We speed up the training using the momentum method [Rumelhart et al., 1986], but otherwise we refrained from more elaborate optimization procedures. In this study, we want to verify the implementation and demonstrate the feasibility of LIF Sampling on BSS-1. In principle, the training could be combined with any optimization

3.3 Experimental setup

Table 3.1: Neuron parameters. Parameters of the network setup specified in ta- ble 3.2. The analog parameters are shown as specified in the software setup and not as realized on the hardware. For details on the calibration procedure see, e.g., [Schmitt et al., 2017]. Legend: ∗the calibration of the membrane time constant was not available at the time of this work, and the corresponding technical param- eter was set to the smallest available value instead (fastest possible membrane dynamics for each neuron). Table taken from Kungl et al. [2019].

A Sampling neuron

Name Value Description

Vreset −35 mV reset potential

Eleak −20 mV resting potential

V_thresh −20 mV threshold potential

Einh −100 mV inhibitory reversal potential Eexc 60 mV excitatory reversal potential

τ_ref 4 ms refractory time

τmem ca. 7 ms membrane time constant∗ Cmem 0.2 nF membrane capacity

τ_synexc 8 ms excitatory synaptic time constant τ_syninh 8 ms inhibitory synaptic time constant

B Bias neuron

Name Value Description

Vreset −30 mV reset potential

Eleak 60 mV resting potential

V_thresh −20 mV threshold potential

Einh −100 mV inhibitory reversal potential Eexc 60 mV excitatory reversal potential

τref 1.5 ms refractory time

τmem ca. 7 ms membrane time constant∗ Cmem 0.2 nF membrane capacity

τ_synexc 5 ms excitatory synaptic time constant τ_syninh 5 ms inhibitory synaptic time constant

C Neurons of the random network

Name Value Description (all analog)

Vreset −60 mV reset potential

E_leak −10 mV resting potential

Vthresh −20 mV threshold potential

Einh −100 mV inhibitory reversal potential Eexc 60 mV excitatory reversal potential

τref 4 ms refractory time

τmem ca. 7 ms membrane time constant∗ Cmem 0.2 nF membrane capacity

τ_synexc 8 ms excitatory synaptic time constant τ_syninh 8 ms inhibitory synaptic time constant

D Synapse

Name Value Description

wbias [0,15] synaptic bias weight in hardware values (digital) w_network [0,15] synaptic network weight in hardware values (digital) d on the order of 1 ms_{(uncalibrated)} synaptic delay, estimated in [Schemmel et al., 2010]

3. Bayesian inference on BSS-1

Table 3.2: Network parameters.Parameters are shown for the three different cases described in the manuscript: (A) Target Boltzmann distribution, Poisson noise. (B) Target Boltzmann distribution, random network for stochasticity. (C) Learning from data, random network for stochasticity. Note that the in-degree, sometimes also referred to as a fan-in factor, represents a neuron’s number of pre-synaptic partners coming from some specific population. Table taken from Kungl et al. [2019].

A Probability distribution with Poisson Noise

Name Value Description

Ns 5 number of sampling neurons N_b 1 number of bias neurons Nr 0 number of random neurons

KRN - within-population in-degree of neurons in the random network Knoise - in-degree of sampling neurons from the random network wRN - synaptic weights in the random network

in hardware units

ν_Poissone/i 300 Hz Poisson frequency to sampling neurons per synapse type

B Probability distribution with random network

Name Value Description

Ns 5 number of sampling neurons N_b 1 number of bias neurons Nr 200 number of random neurons

KRN 20 within-population in-degree of neurons in the random network Knoise 15 in-degree of sampling neurons from the random network wRN 10 synaptic weights in the random network

in hardware units

ν_Poissone/i - Poisson frequency to sampling neurons per synapse type

C High-dimensional dataset

Name Value Description

Ns {207, 208} number of sampling neurons, { rFMNIST, rMNIST } N_b 1 number of bias neurons

Nr 400 number of random neurons

KRN 20 within-population in-degree of neurons in the random network Knoise 15 in-degree of sampling neurons from the random network wRN 10 synaptic weights in the random network

in hardware units

3.4 Experiments and results