Learning Transmission Delays in Spiking Neural

Networks: A Novel Approach to Sequence Learning

Based on Spike Delay Variance

Paul W Wright, Janet Wiles

School of Information Technology and Electrical Engineering

University of Queensland

Brisbane, Queensland 4072, Australia

paul.wright1@uqconnect.edu.au, wiles@itee.uq.edu.au

Abstract— Transmission delays are an inherent component of

spiking neural networks (SNNs) but relatively little is known

about how delays are adapted in biological systems and studies

on computational learning mechanisms have focused on spike-

timing-dependent plasticity (STDP) which adjusts synaptic

weights rather than synaptic delays. We propose a novel algo-

rithm for learning temporal delays in SNNs with Gaussian syn-

apses, which we call spike-delay-variance learning (SDVL). A key

feature of the algorithm is adaptation of the shape (mean and

variance) of the postsynaptic release profiles only, rather than the

conventional STDP approach of adapting the network’s synaptic

weights. The algorithm’s ability to learn temporal input se-

quences was tested in three studies using supervised and unsu-

pervised learning within feed-forward networks. SDVL was able

to successfully classify forty spatiotemporal patterns without

supervision by providing robust, effective adaption of the postsy-

naptic release profiles. The results demonstrate how delay learn-

ing can contribute to the stability of spiking sequences, and that

there is a potential role for adaption of variance as well as mean

values in learning algorithms for spiking neural networks.

spiking neural networks; transmission delays; delay learning;

sequence learning; spike-delay-variance learning; STDP

I. INTRODUCTION

Delays are ubiquitous in the brain. Analog information can be coded within temporal spikes, allowing both static and learnable delays to benefit neural computation in spiking net-works [1]. Real world stimulus analysis is fundamentally a temporal process. Delays play a key role in the dynamics of neuron communication and computation, regularly demon-strated in natural biological systems. For example, barn owls precisely localise the position of fast moving rodents in abso-lute darkness using only their auditory input streams [2]. De-lays created by the magnocellular afferents make it possible for the laminaris neurons to relate phase differences to spatial posi-tion, acting as the required mechanisms for Jeffress style inte-raural coincidence detection [3] [4].

The mechanisms that allow an action potential to affect the membrane potential of another neuron are far from instant. The structural properties of an axon greatly influence the transmis-sion speed such as axon diameter and myelin thickness [5]. The inconsistency of such properties across regions and neuron

types creates variable axonal delay. The synaptic process, where a presynaptic potential gets converted to a postsynaptic potential, takes time. Electrical synapses and chemical syn-apses have different transmission properties, however in gen-eral a postsynaptic potential will have a slower rise time than its presynaptic impulse, causing a synaptic delay in the order of milliseconds [6]. The synapse’s location on the dendritic tree will also create variable electronic transmission delay [7]. All of these delays combine to create an overall transmission delay, which is highly variable between transmission lines but ex-tremely repeatable on the same transmission line. Some ob-served mammalian transmission delays are 1 to 17 ms in Spra-gue-Dawley rats [8], 1 to 32 ms in rabbits [9] and 1 to 30 ms in cats [10].

The ability of a network to produce repeatable temporal spike chains, involving multiple neurons, has been likened to the ability to store and reference memories [11]. Repeatable spike trains require neurons to be capable of consistently re-sponding to a familiar temporal sequence of input spikes. Spike-timing-dependent plasticity (STDP) is a phenomenologi-cal learning rule [12] observed in many real neural networks. STDP represents a family of Hebbian learning kernels that ad-justs the synaptic transmission strength based on the relative timing of the presynaptic and postsynaptic neuron [13]. [ID_1.2] Placed in a neural network containing randomised fixed delays, STDP aids the formation of small scale sequence recognition by fine-tuning weights to reward correlated activ-ity. However, the restriction of fixed delays potentially limits the capabilities of STDP in performing this task.

Although uncommon, there exists some evidence of dy-namic delay learning within biological networks. Variable transmission delays, assumed to be facilitating learning, have been observed in vivo in Mongolian gerbils [14]. Senn, Schnei-der and Ruf suggest that fast delay adaption is unnecessary in the presence of temporal synaptic strength learning rules. They argue that beneficial delays are naturally selected through asymmetric weight adjustment, although at much slower rate [15]. The role of delays and the extent and mechanisms of de-lay learning in mammalian brains is remains unclear.

Computational approaches used to model delay learning vary considerably in their level of biological fidelity. Top-

U.S. Government work not protected by U.S. copyright

WCCI 2012 IEEE World Congress on Computational Intelligence June, 10-15, 2012 - Brisbane, Australia IJCNN

down approaches abstract from the biology details and directly adjust the timing: Supervised learning has been used to directly optimise the delays between internal neurons and output neu-rons in order to achieve the best possible classification per-formance [16]. Another method increases the delay if a postsy-naptic pulse arrives before a defined temporal reference point, and decreases the delay if it arrives after, synchronising multi-ple postsynaptic pulses [17]. Bottom up approaches use more biologically-motivated mechanisms, such as adjusting the con-centration of glutamate receptors in order to vary the perceived transmission delay [18]. An alternative to learning mechanisms which directly adapt delays is to use a network with intrinsic unstable dynamics that are deterministic but vary over a wide range. In such a network, a specific time delayed output can be selected rather than learned. This principle underpins a new approach to temporal networks called liquid state machines [19].

Many computational models have been developed to simu-late spiking neurons such as the Hodgkin-Huxley model [20], the leaky integrate and fire model [21] and the Izhikevich model [22]. They range from high biological fidelity to mathematically abstract, generally with an associated cost in terms of computational speed [23]. Simulation of membrane potentials alone is insufficient to accurately replicate brain ac-tivity due to the huge diversity and complexity of synaptic plas-ticity [24]. Synapses and transmission delays have been mod-elled in a multitude of ways [25].

We focus in this paper on two important features of the synaptic input to a spiking neuron, the temporal variance and the average value. Both interact with the baseline parameters of a neuron to produce a wide range of non-linear behaviour [26]. Analysis of these two features, when a single neuron receives input from a population, reveals intriguing results. Information can be encoded using both features, by either changing the av-erage synaptic input to neuron (corresponding to changing av-erage firing rates of the presynaptic neurons) or by changing the temporal variance of synaptic input (corresponding to the temporal irregularity of the summed postsynaptic potentials). The former cannot produce noticeable network responses at realistic speeds, whereas the latter can [27]. This result sug-gests that temporal synaptic variance is a better medium for quick data transfer and processing than the actual magnitude of input current. Synaptic variance has thus been utilised as a mechanism for effective spatiotemporal sequence classifica-tion, a common task in many biological systems. Barak and Tsodyks adjusted synaptic weights in order to maximise the postsynaptic current variance observed by the integrating neu-ron, yielding successful classification [28]. Two problems with Barak and Tsodyks’ approach to sequence recognition are the insensitivity to temporal reversal of the input pattern and the inability for a neural system to respond more quickly with in-creasing familiarities. Brains are well known for their ability to not only improve accuracy with learning, but also improve the processing speed of known patterns. Such optimisation occurs without reduced robustness.

In the research described in this paper, we aimed to design and test a delay-learning algorithm that could take advantage of synaptic variance as an integral part of adjusting delays. We

designed the algorithm to achieve effective spatiotemporal rec-ognition and classification in small noiseless networks.

The following sections describe our novel transmission de-lay learning algorithm and an associated synaptic transmission model, and demonstrate its performance on sequence tasks in which the order of elements in a sequence changes over time. The learning algorithm is tested in two forms, initially for su-pervised learning and then applied in unsupervised learning tasks. The results demonstrate robust learning of temporal or-der which can adapt over time.

II. METHODS

The model is comprised of key components, including a model neuron (see section II-A), synapse model and learning algorithm (II-B), network and sequence structures (II-C), dif-ference measure and classification success rate (II-D), and learning tasks (II-E).

A. Neuron Model

The model of neuron used in this study was proposed by

Izhikevich as a computationally tractable model of spiking

neurons that balances both simplicity in parameters and com-

plexity in the range of possible neuron behaviours [22]. It is a

part of a set of hybrid dynamical systems that makes use of

dual ordinary differential equations to replicate the complex

neuronal firing patterns such as bursting, threshold variability

and bistabilty [29]. In this paper, ‘regular spiking’ Izhikevich

neurons were used to balance complexity and efficiency, al-

though other neuron models such as the integrate-and-fire

model are capable of performing similar computation. The

first order ordinary differential equations (see [22] for details)

are solved with an improved Euler method, using a time step

of 1ms. Subsequently, all delays and synaptic profiles are

quantised to a resolution of 1ms. Smaller time steps were also

tested and had insignificant impact on the membrane and syn-

aptic dynamics.

B. Synapse Model

In real neural networks, delays are a combination of the

temporal nature of dendritic, axonal and synaptic conduction.

For this study, for the sake of model simplicity and efficiency,

transmission delays were simulated as a single variable acting

in the synapse.

1) Governing Gaussian Function Each synapse releases postsynaptic current into its target

neuron over time which creates conduction delays. Spike de-

lay-variance learning requires a postsynaptic release modelling

equation that has variable delay and temporal delivery width,

for example, the Gaussian function or the sum of rising and

decaying exponentials. A sampled Gaussian function was used

in this paper for computational efficiency. The release of post

synaptic current, I, from a synapse is modelled with a Gaus-

sian function.

(1)

where p is the peak postsynaptic current, t0 is the time since a

presynaptic spike in ms (t0 ≥ 0), μ is the mean and v is the

variance. The parameters p, v and μ are variable for each syn-

apse, and their selection determines the release profile. The

synaptic mean μ represents the peak-timing of postsynaptic

current, such that the transmission delay of the synapse is

equal to μ. Some current will be delivered before and after the

peak timing, depending on the variance. A large variance indi-

cates a wide release profile, whereas a small variance results

in a narrow one. The mean must be non-negative as postsy-

naptic current cannot arrive before the presynaptic spike.

2) Fixed Integral The integral of the profile determines the amount of post-

synaptic current delivered. As a result, the strength (or weight)

of a Gaussian synapse is not dependent on the peak, but by the

integral of the curve. The peak is proportional to the integral,

hence for a given integral value there will only be one corre-

sponding peak value. Globally fixing the integral of all syn-

apses in a network removes the peak as a synaptic variable.

The required peak value for a given variance, mean and global

integral can be computationally determined using an optimisa-

tion algorithm such as gradient descent.

3) The Algorithm The mean and variance of a synapse are only adjusted when

a postsynaptic spike occurs. The temporal error of the synapse is defined as the absolute value of the temporal difference be-tween the synapse’s mean and t0, where t0 is the time difference between the presynaptic and postsynaptic spike. The mean is adjusted to reduce any significant error, helping to replicate the

relative spike timing of the presynaptic and postsynaptic neu-rons. If the error is insignificant but the magnitude of t0 is sig-nificant, the mean is reduced to improve the response time of the postsynaptic neuron. The change of mean, Δμ, is deter-mined by

(2)

where: sgn(.) is the signum function

μ is the mean of the synapse (ms)

k(v) is the learning accelerator

v is the variance of the synapse

ημ is the mean learning rate

α1, α2 are constants

The variance is increased if the synapse’s temporal error is significant, ‘searching’ by broadening the postsynaptic release profile. The variance is decreased if the error is sufficiently small, ‘locking on’ to a familiar temporal spike pattern by nar-rowing the release profile. The change of variance, Δv, is de-termined by

(3)

where: ηv is the variance learning rate β1, β2 are constants

In this paper, μ is restricted to the range [0, 15], v is restricted to the range [0.1, 10] and, for simplicity, k(v) = (v + 0.9)

2 so

that k = 1 at minimum v as the choice of the magnitude of k is arbitrary.

C. Network Structure and Sequence Definition

The networks used for algorithm evaluation comprise of two layers of spiking neurons, an input layer connected to an output layer with a feed-forward structure. Each input neuron i

∈ {1...N} is connected to each output neuron j ∈ {1...M} with a single dynamic Gaussian synapse. The effects of non-uniform network weights and weight adaption are eliminated by enforc-ing that all Gaussian synapses have the same global fixed inte-gral. Each output neuron is connected to all the other output neurons with strong static instant inhibitory synapses, acting as a ‘winner takes all’ mechanism.

During a presentation of a spatiotemporal sequence/pattern, Q input neurons fire exactly once within a given time window, Tm, from a global temporal reference point (the beginning of the presentation). Hence an input sequence is a vector of input

spike times seq = {t1, t2, ... ti} where ti ∈ {0,1...Tm, ∞}, where an infinite spike time denotes silence. If Q is less than the number of input neurons then there is a multivariable compo-nent, which in this paper is referred to as spatial. Supervised learning enforces that the correct output neuron fires at Ts after the presentation of the sequence, unless it has already fired naturally. Thus supervised learning only affects the network if the correct output neuron fails to fire within Ts from the start of a presentation. The transmission delay learning algorithm is

Figure 1. Post synaptic release profiles for variable variance and fixed mean

(top), variable mean and fixed variance (bottom), all restricted by a fixed integral. When the variance is low (0.1), all the current is delivered as a single burst at a

delay determined by the mean. At high variances (>5), current is slowly released

from the synapse over a large period of time, peaking at the delay determined by the mean.

always active whereas supervised learning can be switched on or off during a trial.

D. Sequence Difference Measure and Classification Success

Rate

The task of classifying temporal sequences has variable

difficultly, inversely proportional to the input sequences’ abso-

lute temporal difference. So, in order to quantify algorithm

performance in context of task difficulty, a method of quanti-

fying the temporal difference between two input sequences

needed to be defined. The difference measure, D, between two

normalised sequences is determined by

(4)

where a = (a1, a2, ... aN) and b = (b1, b2, ... bN) are the two

normalised input sequences, Tm is the maximum spike phase

offset. Pattern normalisation is achieved by subtracting the

minimum spike phase offset from all vector elements of a se-

quence, so that at least one term is zero, forcing at least one

input neuron to fire at the beginning of the presentation. Nor-

malisation enforces that two patterns cannot have a large D if

one is a time-shifted version of the other (i.e. temporally iden-

tical). During a classification task, the classification success

rate (CSR) is the percentage of patterns that are represented by

a single unique output neuron, i.e. one particular output neuron

always solely spikes when its corresponding pattern is pre-

sented to the network, but remains silent when all other pat-

terns are presented.

E. Network Tasks

The algorithm was evaluated using three learning tasks, la-belled as Condition 1, 2 and 3.

1) Condition 1. Single Output Neuron Under Condition 1, the circuit had 3 inputs (N = 3), 1 out-

put (M = 1) and 2 patterns, seq a = {0, 3, 7} and seq b = {7, 3,

0} (P = 2, Q = 3). Condition 1 comprised of a single 60 second

trial. Seq a was solely presented during 0-25 and 55-60 sec-

onds, seq b was solely presented during 25-55 seconds. Note

that seq b is a time-reversed version of seq a. Sequences were

presented to the network at a frequency of 2 Hz to allow

membrane potentials to return to baseline before each presen-

tation. Supervised learning was only switched on during 0-15

and 30-45 seconds with Ts = 12ms. The Gaussian synapses

were initialised with v = 2 and μ = 5. The global integral was

fixed at 13 so that at all three postsynaptic pulses are required

to arrive at similar times to induce a spike.

2) Condition 2. Classification of Two Patterns Under Condition 2, the circuit had 10 inputs (N = 10), 2

outputs (M = 2) and 2 patterns (P = 2, Q = 10). Two random input sequences, s1 and s2, are generated at the beginning of each trial with a particular D. During each trial, the network was trained over 100 epochs (an epoch comprises of all pat-terns being presented once to the network at a frequency of 5 Hz) and then the classification success rate (CSR) was calcu-

lated. A trial was only considered successful if the CSR was 100%, otherwise it was considered unsuccessful. The Gaussian synapses were initialised with v ~ U (1, 3) and μ ~ U (2, 10) at the beginning of each trial, where U is the uniform distribution.

Each integer sequence element is an independent random

variable uniformly distributed on [0, Tm], where Tm = 12ms.

The average D between two randomly generated patterns is

0.3563, with a standard deviation of 0.0823. This intuitively

low value is a consequence of pattern normalisation. Thus the

algorithm’s classification performance is evaluated uniformly

on the range D = [0, 0.5], representing the range of sequence

pairs from identical to significantly different. There are 61

possible values of D is this range, as D is a quantised value.

Every value of D is trialled 30 times and the percentage of

successful trials was calculated. Hence there were a total of

1830 uniquely generated experiments, each trial having differ-

ent initial Gaussian parameters and input patterns. The high

number of independent experiments is required to offset any

dependence on initial conditions.

a) Condition 2a. Supervised Learning

Supervised classification activates supervised learning (Ts

= 14ms) during the entire 100 training epochs.

b) Condition 2b. Unsupervised Learning

Unsupervised classification represents a different learning paradigm. There is no teacher to force the output neurons to fire, which is required by SDVL to change synaptic parameters. Initially, a large integral is required, to allow the output neu-rons to spike before the Gaussian synapses are correctly tuned. The integral needs to be reduced over time, so that the synapses can converge to a solution. Integral reduction is a form of simu-lated annealing, and many annealing functions could have been used. Linear annealing was chosen for parameter simplicity. The three linear annealing parameters are: I0 (initial integral), If (final integral) and Tf (time to reach the final integral). The integral starts at I0, linearly descends and reaches If after Tf sec-onds. The integral is then held constant at If for the remaining time.

There are six algorithm parameters to be tuned, ημ, ηv, α1,

α2, β1, β2, in addition to the three simulated annealing parame-

ters. This large parameter space is difficult to manually ex-

plore, thus a genetic algorithm (GA) was used to find solu-

tions. The GA uses a hybrid of mutation and crossover, with a

cost of population diversity. Details of the GA’s implementa-

tion and performance are available in [30].

c) Condition 3. Classification of Forty Patterns

Under Condition 3, the circuit had 40 inputs (N = 40), 40

outputs (M = 40) and 40 patterns (P = 40, Q = 10). Patterns

and Gaussian parameters were initialised as per Condition 2.

SDVL and simulated annealing parameters were evolved as

per Condition 2b. The unsupervised network was trained for

80 epochs. The CSR was calculated after every epoch and

averaged over 20 independent randomly generated trials.

III. RESULTS

A. Condition 1 Results

Under Condition 1, SDVL adapts the synaptic parameters over time to produce effective delay learning (see Fig. 2). Su-pervised learning forced the output neuron to fire with a 12ms phase offset at the beginning of the trial, allowing SDVL to change the synaptic parameters from their initial values. During the first 3 seconds, the synapse connected to input 1 ‘searched’ for the correct delay by spreading out its postsynaptic release profile via an increasing variance. The three means deviate to

find the optimal relative delay configuration such that all post-synaptic peaks arrive at the output neuron simultaneously. The high mean learning rate allowed the means to converge after only 6 seconds. At this point, the delays perfectly matched the input pattern, enabling the variances to reduce to the smallest allowed value (observed as the narrowing of the profiles in Fig. 2 from 10 to 30 seconds). The delays are relatively correct, however are not optimal in terms of response time. SDVL im-proved the response time by reducing the means in a parallel manner, so that their relative timing is conserved. The delays reached the optimal configuration by 80 seconds, indicated by the accelerated output neuron spike phase offset (decreasing

Figure 2. Synaptic parameter traces and neuron spike times during a sequence recognition task. From top to bottom: A) Task design: periods where

supervised learning is active. B) The input and output spike times, shown as a phase offset from the beginning of each sequence presentation. C) Mean traces for all synapses in the network. D) Variance traces for all synapses in the network. E) Gaussian postsynaptic release profiles at various stages throughout the

trial.

from 12ms to 9ms). The input sequence was reversed after 120 seconds, desynchronising the arrival times of the postsynaptic pulses. The output membrane potential was unable to reach threshold due to leak currents in the soma. The unresponsive-ness of the output neuron indicates that the original input pat-tern had been learnt: the output neuron spikes when presented with a known sequence, but remains silent when presented with a different unknown sequence.

Supervised learning was turned back on after 150 seconds (see Fig. 2A), forcing the output neuron to fire, thus changing synaptic parameters. The synaptic means adjusted to match the new input pattern. The variance of the synapse connected to input 3 experienced a huge increase, broadening its profile as the temporal error between its mean and t0 was now large due to the reversed input pattern. Similar to the beginning of the trial, the means converged, and then descended in parallel, which reduces the response time to the optimal value. After 270 seconds, the input sequence was reversed back to the original pattern, rendering the output neuron unable to fire. The unresponsiveness of the output neuron, once again, confirms that the new pattern had been learnt, and the old pattern had been forgotten.

Similar results were observed when replicating the ex-

periment with a reduced global integral of 11. However, the

key difference was the inability of the output neuron to speed

up its response time. Output spikes always occurred with a

12ms phase offset, unable to decrease to 9ms as observed un-

der Condition 1. Increasing the integral to 19 allowed only two

input spikes to induce an output spike. The output spike re-

duced its response time to 5ms, only requiring the first two

spikes in the sequence to fire. The unused synapse diverged to

maximum values. SDVL performs effectively as well when in

the input sequences are distorted with 2ms of uniform tempo-

ral noise (see [30]).

B. Classification Performance

1) Supervised Classification of Two Patterns Under Condition 2a, the average trial success rate was

tested the discrete range of D = [0, 0.5] (see sup in Fig. 3). For

identical sequences (D = 0) classification was always unsuc-

cessful, as it is impossible to uniquely identify two identical

input patterns. As the patterns became more diverse, the suc-

cess rate rapidly increased to over 90% at D = 0.075. The suc-

cess rate reached 100% soon after, when D = 0.16. The aver-

age distance between two random sequences was

0.3563±0.0823 (see Fig. 3). The classification success rate was

perfect within this range, demonstrating that SDVL can accu-

rately classify two arbitrary input sequences when aided by

supervised learning.

2) Unsupervised Classification of Two Patterns The GA consistently found well-performing solutions in the

parameter space. The average trial success rate of a typical

solution (see Table I for parameters) across the entire range of

D under Condition 2b was compared to the supervised case

(see Fig. 3). The success rate was slightly worse than the su-

pervised case, however still could successful classify two arbi-

trary patterns at least 90% of the time, as long as the patterns

had a minimum D of 0.1. Unsupervised classification has a

large dependence on the initial random configurations of the

Gaussian synapses making a 100% trial success rate difficult

to reach for the average range of D. The lack of supervision

caused a slight decrease in performance as suspected, however

effective unsupervised classification is clearly obtainable.

Figure 3. Averaged two pattern classification trial success rate over a wide range of sequence differences for both supervised and unsupervised learning

paradigms.

3) Forty Pattern Unsupervised Performance A well-performing solution was found by the GA (see Ta-

ble I for parameters). The solution’s unsupervised classifica-

tion performance was averaged over 20 trials (see Fig. 4).

SDVL consistently associated each of the 40 output neurons

with a unique pattern, completely characterising the input

space after approximately 20-60 training epochs. SDVL still

performed well even when all input spikes were randomly

time shifted U[0,4] ms at each presentation.

IV. DISCUSSION

A novel transmission delay learning rule, spike-delay-variance learning (SDVL) was proposed and evaluated, con-tributing to our understanding of the range of computations possible with spiking neurons. The algorithm is different from other delay learning rules as it achieves postsynaptic pulse syn-chronisation and response time optimisation in unison. Its sta-bility, reliability and effectiveness for spike sequence recogni-tion and classification were demonstrated in a variety of con-texts. The experiments clearly show that dynamic learnable delays have computational advantages in terms of spatiotempo-ral pattern recognition and classification.

Deterministic spatiotemporal activity patterns appear in real biological systems [31]. Many previous approaches to spatio-temporal spike processing rely on the adaption of weights, ei-ther through combining spike-timing based learning kernels like STDP with variable readout delays [16], or to maximise the combined postsynaptic current variance [28]. SDVL re-solves two issues experienced by Barak and Tsodyks’ [28] ap-proach to variance-based spatiotemporal pattern learning, as it

is sensitive to the temporal direction of the patterns and it can speed up the response speed of a known sequence. A key find-ing of the results is the ability to achieve robust spatiotemporal processing in a network where the weights remain globally uniform. Individual synapse variance and delays are rapidly adjusted to explore the temporal landscape, attempting to achieve synchronised and optimised postsynaptic pulse timing. The results are evidence that varying the timing and shape of a synaptic release profile is computationally advantageous, inde-pendent of strength adaption.

SDVL proved to be a robust classifier of two arbitrary input sequences. The algorithm performs extremely well as a spatio-temporal pattern classifier using both supervised and, the more realistic, unsupervised learning. Without supervision, patterns are required to be slightly more diverse in time, but can still achieve high accuracy and reliability when their temporal dis-similarity remains within a realistic range. The number of pat-terns was increased to 40 without disturbing the effectiveness of SDVL as a classifier. Exploring the algorithm’s parameter space can be a lengthy and tedious process. The usefulness of GA’s to tune the algorithm parameters automatically is an addi-tional result. Sequences were only presented in feed-forward networks once membrane potentials had stabilised and returned to baseline. Disturbing influences present in interconnected networks, such as high network activity and background noise, will force neurons to process postsynaptic potentials when they are initially not at baseline. The consistency of algorithm per-formance potentially could suffer from such influences.

Synapses based on a Gaussian postsynaptic release profile proved to have interesting features. They can widen their re-sponse in order to search for the correct temporal relationship with a presynaptic neuron. Once a beneficial delay is found, they can narrow their response to increase temporal accuracy. If the presynaptic neuron repeatedly fires unpredictably, the release profile gets wider and wider, to a point where the syn-apse has little computation use. Destroying a synapse when it reaches maximum variance and mean would allow integrating neurons to filter out useless spikes, thus only allowing mean-ingful data to reach its soma.

The biological significance of SDVL is yet to be estab-lished. However, there are many examples of biological sys-tems optimising their processing time for frequently encoun-tered input patterns. The algorithm provides a potential method to achieve temporal optimisation of recurrent computation rou-tines via decreased delays. The algorithm could also be applied to nested sequences, providing robust classification over a much longer time scale. To establish a biological significance, empirical studies of dynamic transmission delays are needed.

TABLE I. SIMULATED ANNEALING (IF APPLICABLE) AND SDVL PARAMETERS FOR EACH EXPERIMENTAL CONDITION.

Simulated Annealing Parameters SDVL Parameters

Condition

I0 If Tf ημ ηv α1 α2 β1 β2

1 13 - - 0.03 0.01 3 2 5 5

2a 3 - - 0.025 0.025 3 2 5 3

2b 6.752 4.026 31.4 0.0338 0.0357 1 3 8 5

3 6.754 5.969 514.6 0.0315 0.0218 4 2 1 1

Figure 4. Unsupervised CSR learning curves for forty noiseless and noisy spike

patterns and forty output neurons.

Modern in vivo imaging techniques have increased in accuracy, resolution and duration [32], making such studies achievable. In a recent observation of dynamic transmission delays in vivo in Mongolian gerbils, a phenomenological model was used to estimated that the change of delay per spike was between 5-30 μs [14]. Under Condition 2b and Condition 3, the genetic algo-rithm produced delay learning rates of 31.5 and 33.8 μs per spike respectively (see Table I), similar to the biological esti-mate. However, it must be noted in SDVL the actual change of delay is amplified whenever the synapse’s variance is higher than minimum, according to the learning accelerator in (5). Static delays are clearly important in temporal processing [1], for example, Reichardt detectors [33], hypothesised as a neural mechanism for motion detection in many biological systems, especially insects [34]. It is possible that delays are initially learnt, converging once an optimal configuration is reached. Other learning rules could then perform learning tasks on a shorter time scale, complimented by the delay arrangement.

In conclusion, SDVL’s successful performance suggests that combining dynamic delays with synaptic variance adaption enhances the spatiotemporal processing capabilities of spiking neural networks.

V. FUTURE WORK

Stable transmission delay learning promotes the sustain-

able repeatability of network activity. Izhikevich has proposed

that polychronous neuron groups play a fundamental role in

memory and consciousness [11]. Future work could include

investigating the possible non-random formation of polychro-

nous neuron groups, through robust delay learning. Transmis-

sion delay learning could also be linked with a reward signal.

Neuromodulated delay learning rates would achieve effective

reinforcement learning. Changing the circuit structure to in-

clude memory capacity would allow for serial input of pat-

terns, eliminating the current limitation of one input neuron

per sequence member and enabling more interesting datasets

to be evaluated. The Gaussian integral is considered to be the

weight of a Gaussian synapse. Applying spike-time dependent

plasticity and weight normalisation to the value of the Gaus-

sian integral would dramatically increase the processing po-

tential of the algorithm.

REFERENCES

[1] W. Maass, "On the relevance of time in neural computation and

learning," Theoretical Computer Science, vol. 261, pp. 157-178, 2001.

[2] M. Konishi, "How the Owl Tracks Its Prey: Experiments with trained barn owls reveal how their acute sense of hearing enables them to catch

prey in the dark," American Scientist, vol. 61, pp. 414-424, 1973.

[3] C. Carr and M. Konishi, "A circuit for detection of interaural time differences in the brain stem of the barn owl," The Journal of

Neuroscience, vol. 10, p. 3227, 1990.

[4] C. E. Carr and S. Amagai, "Processing of temporal information in the brain," Advances in Psychology, vol. 115, pp. 27-52, 1996.

[5] S. G. Waxman, "Determinants of conduction velocity in myelinated

nerve fibers," Muscle & nerve, vol. 3, pp. 141-150, 1980. [6] M. V. L. Bennett and R. S. Zukin, "Electrical coupling and neuronal

synchronization in the mammalian brain," Neuron, vol. 41, pp. 495-511,

2004. [7] J. Iles, "The Speed of Passive Dendritic Conduction of Synaptic

Potentials in a Model Motoneurone," Proceedings of the Royal Society of

London. Series B, Biological Sciences, pp. 225-229, 1977.

[8] K. Kandler and E. Friauf, "Development of glycinergic and

glutamatergic synaptic transmission in the auditory brainstem of perinatal rats," The Journal of Neuroscience, vol. 15, p. 6890, 1995.

[9] H. A. Swadlow, "Efferent neurons and suspected interneurons in motor

cortex of the awake rabbit: axonal properties, sensory receptive fields, and subthreshold synaptic inputs," Journal of neurophysiology, vol. 71,

p. 437, 1994.

[10] T. Carney, et al., "A physiological correlate of the Pulfrich effect in cortical neurons of the cat," Vision Research, vol. 29, pp. 155-165, 1989.

[11] E. M. Izhikevich, "Polychronization: Computation with spikes," Neural

Computation, vol. 18, pp. 245-282, 2006. [12] C. Clopath, et al., "Connectivity reflects coding: a model of voltage-

based STDP with homeostasis," Nature neuroscience, vol. 13, pp. 344-

352, 2010. [13] N. Caporale and Y. Dan, "Spike timing-dependent plasticity: a Hebbian

learning rule," Annu. Rev. Neurosci., vol. 31, pp. 25-46, 2008.

[14] S. Tolnai, et al., "Spike transmission delay at the calyx of Held in vivo: rate dependence, phenomenological modeling, and relevance for sound

localization," Journal of neurophysiology, vol. 102, p. 1206, 2009.

[15] W. Senn, et al., "Activity-dependent development of axonal and dendritic delays, or, why synaptic transmission should be unreliable,"

Neural Computation, vol. 14, pp. 583-619, 2002.

[16] H. Paugam-Moisy, et al., "Delay learning and polychronization for reservoir computing," Neurocomputing, vol. 71, pp. 1143-1158, 2008.

[17] H. Hüning, et al., "Synaptic delay learning in pulse-coupled neurons,"

Neural Computation, vol. 10, pp. 555-565, 1998. [18] V. Steuber and D. Willshaw, "A biophysical model of synaptic delay

learning and temporal pattern recognition in a cerebellar Purkinje cell," Journal of computational neuroscience, vol. 17, pp. 149-164, 2004.

[19] W. Maass, et al., "Real-time computing without stable states: A new

framework for neural computation based on perturbations," Neural Computation, vol. 14, pp. 2531-2560, 2002.

[20] A. L. Hodgkin and A. F. Huxley, "A quantitative description of

membrane current and its application to conduction and excitation in nerve," The Journal of physiology, vol. 117, p. 500, 1952.

[21] L. Abbott, "Lapicque's introduction of the integrate-and-fire model

neuron (1907)," Brain Research Bulletin, vol. 50, pp. 303-304, 1999. [22] E. M. Izhikevich, "Simple model of spiking neurons," Neural Networks,

IEEE Transactions on, vol. 14, pp. 1569-1572, 2003.

[23] E. M. Izhikevich, "Which model to use for cortical spiking neurons?," Neural Networks, IEEE Transactions on, vol. 15, pp. 1063-1070, 2004.

[24] L. Abbott and W. G. Regehr, "Synaptic computation," Nature, vol. 431,

pp. 796-803, 2004. [25] R. Brette, et al., "Simulation of networks of spiking neurons: a review of

tools and strategies," Journal of computational neuroscience, vol. 23, pp.

349-398, 2007. [26] J. Pressley and T. W. Troyer, "The dynamics of integrate-and-fire: Mean

versus variance modulations and dependence on baseline parameters,"

Neural Computation, vol. 23, pp. 1234-1247, 2011. [27] G. Silberberg, et al., "Dynamics of population rate codes in ensembles of

neocortical neurons," Journal of neurophysiology, vol. 91, pp. 704-709,

2004. [28] O. Barak and M. Tsodyks, "Recognition by variance: learning rules for

spatiotemporal patterns," Neural Computation, vol. 18, pp. 2343-2358,

2006. [29] E. M. Izhikevich, "Hybrid spiking models," Philosophical Transactions

of the Royal Society A: Mathematical, Physical and Engineering

Sciences, vol. 368, p. 5061, 2010.

[30] P. W. Wright, "Exploring the impact of interneurons and delay learning

on temporal spike computation," Unpublished B.E. Thesis, University of

Queensland, Brisbane, 2011. [31] A. Arieli, et al., "Dynamics of ongoing activity: explanation of the large

variability in evoked cortical responses," Science, vol. 273, p. 1868,

1996. [32] J. N. D. Kerr and W. Denk, "Imaging in vivo: watching the brain in

action," Nature Reviews Neuroscience, vol. 9, pp. 195-205, 2008.

[33] W. Reichardt, Autocorrelation, a principle for the evaluation of sensory information by the central nervous system. New York: Wiley, 1961.

[34] J. Haag, et al., "Fly motion vision is based on Reichardt detectors

regardless of the signal-to-noise ratio," Proceedings of the National Academy of Sciences of the United States of America, vol. 101, p. 16333,

2004.