INHERENTLY STOCHASTIC SPIKING NEURONS FORPROBABILISTIC NEURAL COMPUTATION

Maruan Al-Shedivat1, Rawan Naous1, Emre Neftci2, Gert Cauwenberghs2, and Khaled N. Salama11King Abdullah University of Science and Technology

Thuwal 23955-6900, KSA2Institute for Neural Computation, UC San Diego

La Jolla, CA 92037, USA

AbstractWe propose a new memristive neuron circuit that fol-lows the stochastic spike response model (SRM) andcan be used for spike-based probabilistic algorithms.We show that the switching of the memristor is akin tothe stochastic firing of the SRM. The analysis and sim-ulations confirm that the proposed neuron circuit sat-isfies the neural computability condition that enablesprobabilistic neural sampling and spike-based Bayesianlearning and inference algorithms.

Motivation

• Recent theoretical studies have shown that proba-bilistic spiking can be interpreted as inference andlearning in cortical microcircuits.

• The research on systems that use noise as a computa-tional resource has become a rapidly growing field [1].

• However, such systems have two critical require-ments: (i) the neurons should follow a specific model,and (ii) stochastic spiking should be implemented ef-ficiently for it to be scalable.

• We propose to use the inherent randomness of nano-scale memristors [2] for implementing stochasticallyspiking neurons that fulfill both requirements [3].

Notation

Parameter DescriptionRon and Roff Low and high resistances of the memristorτ0 Average switching time for a device under V = 0

V0 Memristive switching voltage sensitivity

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Stochastic Model of the Memristor

w

D

Anode

Cathode

Insulator(TiO2)

Conductor(TiO2-x)

+

_

+

_

Roff

RonwD

wD

(1- )

Anode

Cathode

The dominantfilament

+

_

Figure 1 : Memristor, the barrier and filament models.

To account for stochastic filament formation, we proposea simple enhanced memristor model:

I = gM

w

D

V, where gMw

D

=Roff − ∆R

w

D

−1

,

dw = f(w,V)dt︸ ︷︷ ︸deterministic term

+(θ(V) ·D−w)dN(τ)︸ ︷︷ ︸stochastic term

,

where the stochastic term follows the dynamics of an in-homogeneous Poisson process dN(τ) with a time con-stant that exponentially depends on applied voltage:

τ(V) = τ0 exp (−V/V0) ,

where τ0 and V0 are parameters of the appropriate units.This leads to the following dynamics:

4 3 2 1 0 1 2 3 4V (volt)

0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.3

0.4

I (m

A)

Voltage Crosses the Threshold

Stochastic memristorDeterministic memristor

4 3 2 1 0 1 2 3 4V (volt)

0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.3

0.4

I (m

A)

Voltage is Under the Threshold

Stochastic memristorDeterministic memristor

Figure 2 : Deterministic and stochastic I-V curves.

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Probabilistically Firing Neuron

Isyn Vreset

Spike shapingand/or

communication

Stochastic Neural Soma

Rm Ron/offCm

outputspikes

Raux

Figure 3 : Memristive neuron circuit.

• Each switching event generated by the memristor isconverted into an analog or a digital spike.

• The circuit was simulated for constant noisy inputsynaptic currents. Spiking statistics and the powerdissipation on the memristor are presented below.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4ISI (sec)

100

101

102

103

Cou

nt

Isyn = 2.2 mA

Isyn = 2.3 mA

Isyn = 2.4 mA

1.8 2.0 2.2 2.4 2.6 2.8Synaptic Current (mA)

0.90

0.95

1.00

1.05

1.10

Coe

ffici

ent o

f Var

iatio

n

Ideal Poisson CVCV of the spikes

Figure 4 : The distribution of inter-spike intervals (ISI).

0 100 200 300 400 500 600 700 800Time (ms)

0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Mem

risto

r Cur

rent

(mA

)

0.1 0.0 0.1 0.2 0.3 0.4Memristor Current (mA)

100

101

102

103

104

105

Tim

e (m

s) < 3% of the total time

Figure 5 : Current through the memristor during spiking.

Spiking Winner-Take-All Network

YXi

YXj

Population Coding

WTA Layer

LateralInhibition

Synaptic weightswith STDP learning

Z1

Z2

ZK

0.0 0.2 0.4 0.6 0.8

Time (sec)

5

10

15

20

25

30

Out

put n

euro

n in

dex 0

1

2

3

4

Figure 6 : Synaptic weights and WTA spiking activity.

Parameter name Parameter value & classification accuracyOutput layer size 16 32 64 128Accuracy 58.9% 64.2% 73.9% 78.4%Robustness to memristor imperfections (32 output neurons)

Variability in τ0 0% 20% 40% 60%Accuracy 64.3% 64.2% 62.5% 63.0%Variability in V0 5% 10% 15% 20%Accuracy 54.1% 42.0% 27.8% 16.6%

Table 1 : WTA Performance Under Different Conditions.

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Spiking Boltzmann Machine

Spike encoding of thevisual pattern

Spike encodingof the labels

Plasticsynapses

Hidden Layer

Visible Layer

[Every non-zero pixel and label drives the corresponding encoding neuron with positive injected current]

We trained an RBM and mapped its parameters onto aspiking neural network consisting of 824 visible units and500 hidden units. The same network was capable of bothdiscrimination (classification) and generation:

Figure 7 : Raster plot of the visible layer spiking activity inthe neuromorphic RBM (classification and restricted re-construction tasks) and the reconstructed visual patterns.

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Discussion and Future Work

• The proposed memristive stochastic spiking neuronis more area- and power-efficient than the classicalnoise injection approach, and hence is scalable.

• Future work includes large-scale circuit-level simula-tions and fully memristive spiking system design.

Conclusions• The proposed neuron is a native implementation of

the stochastically firing SRM [4].• The proposed noise generation mechanism based

on memristive switching is power-efficient.• Full compatibility with WTA and RBM networks.

References[1] W. Maass, “Noise as a resource for computation and learning in

networks of spiking neurons,” Proceedings of the IEEE,vol. 102, pp. 860–880, May 2014.

[2] S. Gaba et al., “Stochastic memristive devices for computingand neuromorphic applications,” Nanoscale, 2013.

[3] M. Al-Shedivat et al., “Memristors empower spiking neuronswith stochasticity,” (submitted).

[4] R. Jolivet et al., “Predicting spike timing of neocorticalpyramidal neurons by simple threshold models,” Journal ofcomputational neuroscience, vol. 21, no. 1, pp. 35–49, 2006.

7th International IEEE EMBS Neural Engineering Conference, Montpellier, France, April, 2015 maruan.shedivat@kaust.edu.sa