An Efficient Method for online Detection of Polychronous Patterns in Spiking Neural Network

TL;DR

This paper introduces a novel online detection method for polychronous patterns in spiking neural networks, significantly improving efficiency and enabling real-time recognition of spike-timing sequences.

Contribution

The work presents a new model that captures polychronous patterns directly during neural simulation using randomized codes and hash tables, enabling online detection.

Findings

Method outperforms existing detection techniques in computational efficiency.

Successfully recognizes spike-timing patterns in a visual task.

Enables real-time pattern detection in neural simulations.

Abstract

Polychronous neural groups are effective structures for the recognition of precise spike-timing patterns but the detection method is an inefficient multi-stage brute force process that works off-line on pre-recorded simulation data. This work presents a new model of polychronous patterns that can capture precise sequences of spikes directly in the neural simulation. In this scheme, each neuron is assigned a randomized code that is used to tag the post-synaptic neurons whenever a spike is transmitted. This creates a polychronous code that preserves the order of pre-synaptic activity and can be registered in a hash table when the post-synaptic neuron spikes. A polychronous code is a sub-component of a polychronous group that will occur, along with others, when the group is active. We demonstrate the representational and pattern recognition ability of polychronous codes on a direction selective visual task involving moving bars that is typical of a computation performed by simple cells in the cortex. The computational efficiency of the proposed algorithm far exceeds existing polychronous group detection methods and is well suited for online detection.

Equations

$$v^{\prime }=0.04v^2+5v+140-u+I$$

$$u^{\prime }=a(bv-u)$$

$$\text{if}v>30mV\text{then}\{\begin{array}{l}v\leftarrow c\\ u\leftarrow u+d\end{array}$$

Full text

††journal: Neurocomputing

1 Introduction

Spiking neurons [1] present quite a different paradigm to those of artificial neural networks that work directly on real valued variables [2]. Often, investigators choose to decode the spiking activity using various methods [1] into real values such that they can be used with traditional regression and classification algorithms [3].

Polychronization [4] is a spiking model of memory and computation that avoids this artificial decoding process. Instead, it treats each causally bound cascade of spiking activity as a distinct memory or computation. Precisely repeating spatio-temporal patterns of spikes, and the underlying network structures (defined by connections, axon delays and synaptic weights) that facilitate them, are defined as Polychronous Neural Groups (PNGs). In addition to avoiding arbitrary decoding of spiking signals, PNGs also have the advantage of linking up with a number of seminal theories in neuroscience. The activation of PNGs reflect the Pattern Recognition Theory of Mind [5] in which all mental content and computation is reduced to the combination of a large number of pattern recognizers. When we consider the synaptic adaptation that is required to form the neural structure of PNGs, Hebbian Cell Assembly [6] is descriptive of the organizational process of neural representation. If we also consider the synaptic adaptation as occurring in a competitive environment within the brain, Neural Darwinism (or the Theory of Neuronal Group Selection) [7, 8] also becomes significant in describing this formative process. There are recent computational neuroscience works that have used PNGs to study cognitive representation [9, 10, 11, 12, 13], as well as the basis for a model of working memory [14, 15, 16]. In general, the PNG model of spiking computation presents the potential for a significantly higher capacity [4, 17, 18, 19] over previous forms of spike-coding.

Why then are PNGs not more widely employed in the application and study of spiking neural models? We suggest two reasons. Firstly, that it is enticing to integrate spiking models with machine learning methods instead, due to the clearer mathematical underpinnings of that field, as well as the recent advances and generated interest [20]. Secondly, the algorithms currently available to detect PNGs [4, 21] are inefficient and typically cannot be run on-line, but rather on stored spiking data.

The aim of this work is to address this second limitation by exploring a minimal model of polychronization that can be used to detect precisely recurring temporal patterns of spiking activity in a highly efficient manner that can execute as part of the spiking simulation and therefore run in an on-line fashion. It is hoped that this will contribute to a vein of work [22, 23, 24, 25] that is currently attempting to facilitate the study and application of spiking networks in their own terms, rather than resorting to more general machine learning frameworks.

1.1 Polychronous Neural Groups

A PNG [4] is a time-locked pattern of activity that cascades through a set of neurons. Apart from the external input required to trigger the PNG, no further input is required to cause all constituent neurons to fire at their precise time. The structure of a PNG is defined at three levels, each of which are visualized in Figure 1. The potential for a PNG is determined by its structure in terms of connectivity and conduction delays: these determine the possibility of pre-synaptic action potentials (spikes) to arrive simultaneously and thus cause further spikes. The adapted synaptic weights at these crucial junctures must be strong enough to propagate enough current to activate the PNG. Finally, the external input to the neural circuit must match the triggering anchor neurons in order for any PNG to become active during simulation. Each level of a PNGs definition is dependent upon the former.

Originally, PNGs were introduced as a potential substrate for the neural groups in the theory of Neural Darwinism [7, 8, 4]. From the outset, they have been demonstrated to have extremely high computational/memory capacity both in theory [4] and in practice [17, 18, 19]. Since their introduction, PNG’s have also been applied to pattern recognition tasks in both supervised [26, 27] and unsupervised [28, 29, 30, 31] forms. The main benefit of using a polychronous representation like a PNG is that it retains all of the spatio-temporal activity within a spiking network without the need to convert the activity to another representation that inevitably loses much of the information.

1.2 PNG Detection Algorithms

Initial methods proposed for the detection of PNGs [4, 21] were predominantly brute-force approaches that tested every combination of input stimulus that was possible to trigger a PNG activation. This would be done in a two stage process. Firstly, adapted PNGs would be determined by optimally stimulating every set of three group triggering neurons and record the resulting activity for each combination. Secondly, activated PNGs would be detected in the spiking activity by pattern matching each triplet of spikes against the stored adapted PNGs. The inefficiency of these procedures prohibited their wide application. More recently, alternative methods have emerged that improve the efficiency of detecting active PNGs [24] or a probabilistic fingerprint of polychronous activity [22].

1.3 Motivation for a Polychronous State

From their introduction, the general concepts of Polychronous activity and Polychronization of neural networks has been distinct from the specific structure of a PNG [4]. The latter were used to explore the structural nature of spatio temporal neural activity as well as a conceptual link to Neural Darwinism and the Theory of Neural Groups Selection.

It should be noted that in many respects, PNGs have an arbitrary definition and one that comes with a few restricting limitations:

PNGs must be triggered by precisely three anchor neurons connected through a single root neuron. 2. 2.

The minimum network path length of a PNG must be seven or other arbitrary number. 3. 3.

Identification of a PNG must happen in a silent, noiseless network. 4. 4.

Network boundaries for a PNG are fuzzy, they must be truncated for reliable active detection.

While PNGs have their role for structural and network capacity analysis, when the task is real-time pattern recognition, the disadvantages outweigh the benefits of using them. For this use-case, a method is needed for quantifying the polychronous state of a network at any point in time during the presentation of a pattern, or at the end. In the section that follows, we introduce an algorithm to form a polychronous encoding during the computation of neural spiking activity that can fill this role.

2 A Minimal Polychronous Model

We simplify the requirements for a polychronous pattern in a number of ways. Firstly, the atomic unit of polychronization is defined to be a single spike, rather than a groups of neurons. Getting rid of the group structure also rids us of the arbitrary boundary conditions that determine the neurons within the group, i.e. precisely three triggering anchor neurons and a lower threshold on the maximum path length of the PNG. Secondly, a polychronous pattern is solely based on neural activity, not on the structure of the network or synaptic strengths. This removes the dependency of searching for structural and dynamical PNGs before detecting activated ones.

We observe that during the activation of a PNG, the pre-synaptic sequence of spikes will be fixed for each generated spike, otherwise it would constitute a different PNG. Hence, we define a generated spike with a fixed order of pre-synaptic input spikes to be a distinct polychronous pattern. This is illustrated in Figure 2. In our proposed scheme, each ordering of pre-synaptic spikes produces a different code that is unique to a polychronous pattern. This process is described in the next section on detection. As the generation of these codes is central to our method, we refer to these minimal polychronous patterns as polycodes in order to distinctly identify them from PNGs.

We can say for certain that a particular polycode will always activate when a particular PNG activates. Therefore, if PNG X is active whenever stimulus X is presented, then polycode X will also be active. Of course, polycode X has the potential to activate when PNG X does not. This logic means that polychronous codes have the same response consistency properties of PNGs but that individual polycodes are not guaranteed to be as representationally selective. Thus, a polycode is a sub-component of a PNG.

Figure 3 illustrates the formation of a polycode and its subsequent activation when the post-synaptic neuron spikes. The algorithm for the tagging and bit rotation parts are thoroughly explained in the methods section.

The theoretical capacity of polycodes in a given network is $(N\cdot S!)$ where $N$ is the number of neurons and $S$ the number of synapses per neuron. Due to the vast capacity for any networks with more than about ten synapses per neuron, the bit precision of the polycodes are the limiting factor. Depending on whether a 32 bit or 64 bit code is selected, the capacity would be about 4 billion or 18 billion billion, respectively. The chances of polycode collisions within this space are determined by the hash spread function and the number of polycodes that occur in a given spiking network. In our experiments, explained in later sections, there are about half a million polycodes observed which falls within an acceptable range to avoid collisions with either bit precision.

3 Methods

3.1 Neural Network

The neural network model used in this work follows the implementation defined in [4]. Recurrently connected neurons, denoted by $L$ are stimulated by the inputs directly as injected current, $I$, that perturbs the membrane potential modeled with a simple model [32]. This method for modeling the spiking activity of a neuron is shown to reproduce most naturally occurring patterns of activity [33]. The real-valued inputs are normalized between [math] and $1$, which are multiplied by a scaling factor of $20$ before being injected as current into $L$. Input connections project from a $16x16$ grid of pixels, each stimulating a single excitatory neuron. The network activity dynamics are then simulated for $30ms$.

For our experiments the network consists of 320 spiking neurons with the ratio of excitatory to inhibitory as 256:64. Neurons are pulse-coupled with static synapses i.e. the delta impulse (step) function. Connectivity is formed by having $N^{2}\cdot C$ synapses that each have source and target neurons drawn according to uniform random distribution, where $N$ is the number of neurons and $C$ is the probability of a connection between any two neurons. Weights are drawn from two Gaussian distributions; $\mathcal{\mathcal{{N}}}(6,0.5)$ for excitatory and $\mathcal{{N}}(-5,0.5)$ for inhibitory. All parameters for excitatory and inhibitory neuron membranes are taken from [32]. The equations for the membrane model are as follows:

[TABLE]

[TABLE]

With the spike firing condition:

[TABLE]

3.2 Task and Stimuli

A simple visual task to determine direction selectivity of motion is taken from a recent study in computational neuroscience [34]. This task is suitable for the small, cortical column sized [35] network that we are working with that is connected directly to the visual stimuli – i.e. low in the cortical hierarchy. The inputs consist of moving bars that take one of eight directions, 0*∘, in 45∘* increments, through to 315*∘*. Static images of these input patterns are visualized in Figure 4. The frame dimensions are $16x16$ pixels, each one is used as an exclusive input to a single excitatory neuron. This direction selectivity task is used in later sections to establish the representational ability of polycodes and an example of their use in pattern recognition.

3.3 Polychronous Pattern Detection

Pseudo code that describes the algorithm to generate polychronous codes is given as follows:

// to be called every simulation time-stepif ($this$ neuron spikes) {for each post-synaptic neuron $post$ {XOR($post$.code, $this$.tag)ROTL($post$.code, 1)}// ignore spikes caused by external inputif ($this$.code != $this$.tag) {$hashmap$.emplace($this$.code, $label$)// reset code$this$.code = $this$.tag}}// only combine causal activity in the codeif ($this$.Vm < 0) {$this$.code = $this$.tag}

This code can be integrated with the core of a time-step based spiking neural network simulation. The $tag$ values are randomly generated binary strings that are fixed for each neuron in the network. The $code$ values are the polychronous codes that are initialized per neuron, as the $tag$ and subsequently updated according to the pseudo code.

On the occurrence of a spike, two things are triggered. Firstly, all of the codes at the post-synaptic neurons are XOR’d with the pre-synaptic neurons tag code and their bits are rotated. This is the step that generates evenly spread and likely unique valued codes for each combination of pre-synaptic activity that causes a spike. Secondly, the polychronous code value for the neuron that has just spiked is used as a hash key in a lookup table. This should only occur if the code is different from its initial value, otherwise the spike will have just been caused by external input. Information about the pattern can be stored in the cell, such as a class label or a repetition value. The last part of the pseudo code is run every time the neuron membrane activity is updated. It resets the polychronous code to its initial value if the membrane potential crosses a lower threshold so that the code only reflects pre-synaptic activity that had a causal role in generating a spike.

4 Results

4.1 Stability of Repeating Patterns

The repeatability of patterns are the fundamentally required property for them to form representations of input stimuli [9, 10]. Initially, all patterns will be newly registering and it will take time for repeats to occur. Figure 5 plots the occurrence of novel and repeating polycodes while a directional stimulus is presented over 100 seconds. Each point in the graph is an average of the eight input stimuli.

In each second of simulation, there are about $15k$ polycodes active. That corresponds to a neural network activity level of just under $5\%$ on each millisecond time-step. The number of repeating polycodes overtakes the number of novel ones at the six second mark. Eventually, there are over $10k$ repeating polycodes, which have the potential for representational consistency. The remaining level of $3.5k$ novel polycodes indicates there is continual source of new patterns in the neural activity, given that the input patterns are uniformly repeating. This continual occurrence of new patterns must be due to the repeating inputs convolving with the fading memory of the spiking activity.

4.2 Representational Selectivity

Activation consistency alone is not a sufficient condition for a representational system. Polycodes also must be shown to be selective, i.e. are only active when a subset of the input types are presented, ideally a single type of input sample. Figure 6 plots the number of polycodes that are active for each quantity of input sample direction.

The bulk of the polycodes are active when two directions of input are presented as stimuli. However, there are a significant number, above $100k$ in total, that are only active for a single particular direction. Also, there are comparatively few polycodes that are active for any direction which indicates that the coding method is highly sensitive to the input stimuli.

4.3 Pattern Recognition

The previous properties of polycode occurrence, consistency and selectivity are now utilized in a pattern recognizer.

During the training phase, 100 second sample of each directional input is presented. Whenever a polycode is active, two values are stored at the corresponding hash table cell: ${directionLabel,repeats}$. Upon a hash table lookup $repeats$ is incremented if $directionLabel$ matches and is decremented otherwise. If $repeats$ goes down to zero, the $directionLabel$ switches to the current sample’s direction. In the classification phase, an $nDirection$ dimension prediction vector, $pred$, is formed in which $pred[directionLabel]=\sum log_{2}(repeats)$. Finally, the predicted direction is determined by $max(pred(\cdot))$. Figure 7 plots the prediction vector for each of the presented samples (along the y-axis) with the $log_{2}(repeats)$ values for each $directionLabel$ (along the x-axis).

This simple pattern recognition method manages to amplify the effect of the polycodes that repeat in response to particular patterns and thus forms the basis of an effective classifier for this low-level visual task.

4.4 Efficiency

The detection of minimal polychronous patterns as proposed in this article imposes an overhead throughout the spiking simulation, instead of running as an off-line process that scans through spike data generated by the simulation. Whenever a spike occurs, a few extra instructions must execute per synapse along with a single hash table lookup.

The efficiency of the proposed algorithm cannot be directly compared to traditional PNG detection because the patterns detected by it are substantially simplified when compared to the structural and temporal information contained implicitly within a PNG. However, for a reference, it takes about 23 minutes to perform one pass of PNG detection using the code distributed along with the introductory paper of polychronization [4]. This stands in contrast to the 39ms overhead per ten simulated seconds imposed by our minimal polychronous pattern detection. A comparison of the runtime efficiency between PNG and polycode detection is shown in Figure 8. The overhead of polycode detection can be seen in the right hand plot and is four orders of magnitude smaller than PNG detection time shown on the left.

5 Discussion

5.1 Advantages

The detection of polychronous codes provides a rapid way to detect precise spike-timing patterns. Previously, there was a choice: inefficiently detect PNGs [21], decode spike sequences into real-values [1], or perform some computation of distance between the spike sequences themselves [36]. The latter two options do not have the ability to reliably distinguish spatio-temporal spiking patterns from their output values. This minimal method of polychronous code detection is even more efficient than recent alternative forms of PNG detection [22, 24], which are themselves vast improvements over the initial algorithms. The fastest of these alternative methods [22] requires several hundred extra seconds of spiking simulation per stimuli in order to detect the equivalent of a polychronous response.

We have shown through a simple visual motion sensitivity task that polycodes have the properties of response consistency and selectivity as required by representational systems. These properties have been exploited in the construction of a pattern recognizer that works on polycodes directly. The minimal model of polychronization described here also has the advantage of removing many arbitrary constraints on the definition of a PNG. A polychronous pattern need no longer require triggering by precisely three anchor neurons. Also, the arbitrary threshold imposed by a minimum longest network path length is not present.

5.2 Limitations

The minimal model of polychronization proposed has none of the structural information that PNGs contain implicitly. This is a particularly serious limitation if the intent is to analyse structural properties of a network through detected PNGs. However, the works using polychronization to date have largely used PNGs as a representational model that only relies upon their formation and occurrence in response to input stimuli [27, 11], not their structural properties.

Another limitation is the theoretically reduced selectivity of polycodes as compared with PNGs. By definition, the polycodes have the same or better response consistency of PNGs but this does not hold with selectivity. In fact, it is very likely that polycodes are far less selective than PNGs due to their far simpler activation requirements. This problem would need to be mitigated by building a representational system around populations of polycodes instead of a paradigm of $1\ class=1\ code$.

5.3 Future Work

The model and methods outlined in this work is just the basis of a simpler, more efficient form of polychronization. There are a number of key areas that are in need of investigation using this new methodology.

Plasticity forming representations.

Our minimal model of polychronization can be applied to any spiking network activity, unlike the original model, which relied upon the evolution of synaptic weights through STDP [4]. However, plasticity has a central role in the functional self-organization of the nervous system in response to environmental stimuli. Therefore, it is essential to investigate the emergence of polychronous codes in response to specific input patterns while the synapses are adapting according to plasticity. In particular, we stress the importance of analysing the representational properties of these codes to determine if unsupervised synaptic adaptation can improve their response consistency and selectivity.

Hierarchical polychronous patterns.

The experiments presented here use a single recurrently connected network to obtain a polychronous response from the input stimuli. This is analogous to a single cortical mini-column [35] that might be detecting one type of pattern in the mammalian brain. For a truly powerful representational system, it is expected that pattern recognizers work in a massive hierarchy in which higher levels respond to increasingly abstract features of the input [5]. In terms of experimentation, the response consistency and selectivity of polychronous codes could be measured for a series of connected layers of networks which each use the previous networks output as its own input. It would be expected that consistency and selectivity increases with additional layers. This would indicate a higher degree of invariance as well as the ability to recognize higher level patterns, more general than localized spatio-temporal patterns.

Regression using polychronization.

The representative nature of PNGs and polychronous codes make them particularly suitable for classification tasks. Regression problems generally require a quantitative output that can be combined with trainable real-valued parameters in order to approximate a desired signal. We take inspiration from the Cerebellar Model Articulation Controller (CMAC) from autonomous robotics [37].This model is arguably not a network at all, but rather combines sensor input with internal state and maps the result to a set of cells through a hash table. The CMAC model enables regression by storing a real-value at each cell of its hash table. Activated cell values are trained by iterative gradient descent. We propose that it is possible to use this regressive model when the hash table cells are determined with polychronous codes, thus enabling function approximation in addition to pattern recognition that is typically associated with polychronization.

It is hoped that the simple approach and algorithm presented in this paper can facilitate investigations to the above areas as well as others that the authors cannot foresee.

Acknowledgments

This work was supported in part by an Engineering and Physical Sciences Research Council’s Doctoral Training Grant through University of Surrey.