Towards Spatio-Temporal Pattern Recognition Using Evolving Spiking Neural Networks Stefan Schliebs, Nuttapod Nuntalid, and Nikola Kasabov Auckland University of Technology, KEDRI, New Zealand
[email protected] www.kedri.info
Abstract. An extension of an evolving spiking neural network (eSNN) is proposed that enables the method to process spatio-temporal information. In this extension, an additional layer is added to the network architecture that transforms a spatio-temporal input pattern into a single intermediate high-dimensional network state which in turn is mapped into a desired class label using a fast one-pass learning algorithm. The intermediate state is represented by a novel probabilistic reservoir computing approach in which a stochastic neural model introduces a non-deterministic component into a liquid state machine. A proof of concept is presented demonstrating an improved separation capability of the reservoir and consequently its suitability for an eSNN extension.
1 Introduction The desire to better understand the remarkable information processing capabilities of the mammalian brain has recently led to the development of more complex and biologically plausible connectionist models, namely spiking neural networks (SNN). See e.g. [4] for a comprehensive standard text on the material. These models use trains of spikes as internal information representation rather than continuous variables. Nowadays, many studies attempt to use SNN for practical applications, some of them demonstrating very promising results in solving complex real world problems. An evolving spiking neural network (eSNN) architecture was proposed in [16]. The eSNN belongs to the family of Evolving Connectionist Systems (ECoS), which was first introduced in [7]. ECoS based methods represent a class of constructive ANN algorithms that modify both the structure and connection weights of the network as part of the training process. Due to the evolving nature of the network and the employed fast one-pass learning algorithm, the method is able to accumulate information as it becomes available, without the requirement of retraining the network with previously presented training data. The eSNN classifier learns the mapping from a single data vector to a specified class label. This behavior is very suitable for the classification of time-invariant data. However, many of today’s data volumes are continuously updated adding an additional time dimension to the data sets. The classification of spatio-temporal patterns is a great challenge for data mining methods. Many data vectors are sequentially presented to an algorithm which in turn learns the mapping of this sequence to a given class label. In its current form, eSNN does not allow the classification of spatio-temporal data. K.W. Wong et al. (Eds.): ICONIP 2010, Part I, LNCS 6443, pp. 163–170, 2010. c Springer-Verlag Berlin Heidelberg 2010
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In this paper an extension of eSNN is proposed that enables the method to process spatio-temporal information. The general idea is to add an additional layer to the network architecture that transforms the spatio-temporal input pattern into a single highdimensional network state. The mapping of this intermediate state into a desired class label can then be learned by the one-pass learning algorithm of eSNN. We present here an initial experimental analysis demonstrating the feasibility of the proposed extension in principle and leave a more comprehensive analysis for future studies.
2 Extending eSNN for Spatio-Temporal Pattern Recognition The eSNN classification method is built upon a simplified integrate-and-fire neural model first introduced in [14] that mimics the information processing of the human eye. We refer to [12,11] for a comprehensive description and analysis of the method. The proposed extension of eSNN is illustrated in Figure 1. The novel parts in the architecture are indicated by the highlighted boxes. We outline the working of the method by explaining the diagram from left to right.
Fig. 1. Architecture of the extended eSNN capable for processing spatio-temporal data. The dashed boxes indicate novel parts in the original eSNN architecture.
Spatio-temporal data patterns are presented to the system in form of an ordered sequence of real-valued data vectors. In the first step, each real-value of a data vector is transformed into a spike train using a population encoding. This encoding distributes a single input value to multiple neurons. Our implementation is based on arrays of receptive fields as described in [1]. Receptive fields allow the encoding of continuous values by using a collection of neurons with overlapping sensitivity profiles. In [11] the role of the encoding was investigated and suitable parameter configurations were suggested. As a result of the encoding, input neurons spike at predefined times according to the presented data vectors. The input spike trains are then fed into a spatio-temporal filter which accumulates the temporal information of all input signals into a single highdimensional intermediate state. We elaborate on the specifics of this filter in the next section. The one-pass learning algorithm of eSNN is able to learn the mapping of the
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intermediate state into a desired class label. The learning process successively creates a repository of trained output neurons during the presentation of training samples. For each training sample a new neuron is trained and then compared to the ones already stored in the repository. If a trained neuron is considered to be too similar (in terms of its weight vector) to the ones in the repository (according to a specified similarity threshold), the neuron will be merged with the most similar one. Otherwise the trained neuron is added to the repository as a new output neuron. The merging is implemented as the (running) average of the connection weights, and the (running) average of the two firing threshold. We refer to [12] for a more detailed description of the employed learning in eSNN and to [11] for a comprehensive analysis of all parameters involved in the training process. In this study, the implementation of the spatio-temporal filter employs concepts of the reservoir computing paradigm [15]. The reservoir is represented by a large recurrent neural network whose topology and connection weight matrix is fixed during simulation. The function of the reservoir is to project the network inputs into a highdimensional space in order to enhance their separability. Then, an external classification method, i.e. a readout function, maps the reservoir state into a class label. Due to the use of recurrent networks, the state of the reservoir can incorporate temporal information present in the input signals. Thus, the reservoir approach is very suitable to process spatio-temporal data. A reservoir based on leaky integrate-and-fire (LIF) neurons was proposed in [9] and is commonly referred to as a Liquid State Machine (LSM). LSMs have attracted a lot of research interest; see [13] for a review. In [3,17] it was argued that the LSM approach is biologically very plausible, since some parts of the mammalian brain might act as a liquid generator while other brain areas learn how to interpret the perturbations of the liquid caused by external sensory stimuli. From this viewpoint, LSM models mimic brain-like information processing and their analysis might not only lead to very powerful computational tools, but may also provide further insights into the functioning of the mammalian brain. 2.1 A Spatio-Temporal Filter Based on a Probabilistic Liquid State Machine For the extension of eSNN, we propose a probabilistic LSM which replaces the deterministic LIF neurons of a traditional LSM with stochastic neural models. This approach is motivated by the fact that biological neurons exhibit significant stochastic characteristics which, we believe, have to be taken into account when modelling a brain-like information processing system. Probabilistic neural models have been proposed in many studies, e.g. in the form of dynamic synapses [10], the stochastic integration of the postsynaptic potential [4] and stochastic firing thresholds [2], but also in [8] where the spike propagation and generation are defined as stochastic processes. As an initial study, we employ some very simple probabilistic extensions of the LIF model in the probabilistic LSM. These stochastic models are well-known and are comprehensively described in [4]. Based on a brief summary of the LIF neural model, we explain the probabilistic extensions in the next paragraphs. The LIF neuron is arguably the best known model for simulating spiking networks. It is based on the idea of an electrical circuit containing a capacitor with capacitance C
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and a resistor with a resistance R, where both C and R are assumed to be constant. The model dynamics are then described by the following differential equation: τm
du = −u(t) + R I(t) dt
(1)
The constant τm is called the membrane time constant of the neuron. Whenever the membrane potential u crosses a threshold ϑ from below, the neuron fires a spike and its potential is reset to a resting potential ur . It is noteworthy that the shape of the spike itself is not explicitly described in the traditional LIF model. Only the firing times are considered to be relevant. We define a stochastic reset (SR) model that replaces the deterministic reset of the potential after spike generation with a stochastic one. Let t(f ) : u(t(f ) ) = ϑ be the firing time of a LIF neuron, then lim
t→t(f ) ,t>t(f )
u(t) = N (ur , σSR )
(2)
defines the reset of the post-synaptic potential. N (μ, σ) is a Gaussian distributed random variable with mean μ and standard deviation σ. Variable σSR represents a parameter of the model. We define two stochastic threshold models that replace the constant firing threshold ϑ of the LIF model with a stochastic one. Once more, let t(f ) be the firing time of a LIF neuron. In the step-wise stochastic threshold (ST) model, the dynamics of the threshold update are defined as lim ϑ(t) = N (ϑ0 , σST ) (3) t→t(f ) ,t>t(f )
Variable σST represents the standard deviation of ϑ(t) and is a parameter of the model. According to Eq. 3, the threshold is the outcome of a ϑ0 -centered Gaussian random variable which is sampled whenever the neuron fires. We note that this model does not allow spontaneous spike activity. More specifically, the neuron can only spike at time t(f ) when also receiving a pre-synaptic input spike at t(f ) . Without such a stimulus a spike output is not possible. The continuous stochastic threshold (CT) model updates the threshold ϑ(t) continuously over time. Consequently, this model allows spontaneous spike activity, i.e. a neuron may spike at time t(f ) even in the absence of a pre-synaptic input spike at t(f ) . The threshold is defined as an Ornstein-Uhlenbeck process [6]: τϑ
√ dϑ = ϑ0 − ϑ(t) + σCT 2τϑ ξ(t) dt
(4)
where the noise term ξ corresponds to Gaussian white noise with zero mean and unit standard deviation. Variable σCT represents the standard deviation of the fluctuations of ϑ(t) and is a parameter of the model. We note that ϑ(t) has an overall drift to a mean value ϑ0 , i.e. ϑ(t) reverts to ϑ0 exponentially with rate τϑ , the magnitude being in direct proportion to the distance ϑ0 − ϑ(t). The following parameters were used to configure the neural models. We set the membrane time constant τm = 10ms, the resting potential ur = 0mV, the firing threshold
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Stimulus Standard LIF Step-wise noisy threshold Noisy reset Continuous noisy threshold time Fig. 2. Evolution of the post-synaptic potential u(t) and the firing threshold ϑ(t) over time (blue (dark) and yellow (light) curves respectively) recorded from a single neuron of each neural model. The input stimulus for each neuron is shown at the top of the diagram. The output spikes of each neuron are shown as thick vertical lines above the corresponding threshold curve.
ϑ0 = 10mV, the after-spike refractory period Δabs = 5ms, the standard deviation of reset fluctuations σSR = 3mV, the standard deviation of step-wise firing threshold σST = 2mV and the standard deviation of continuous firing threshold σCT = 1mV. The dynamics of the four models are presented in Figure 2. For each model a single neuron is shown that is stimulated by a random spike train generated by a Poisson process with mean rate 150Hz. Both the evolution of the post-synaptic potential u(t) and the evolution of the firing threshold ϑ(t) are recorded and shown in the figure. We note the step-wise and the continuous update of the two threshold models and the stochastic reset of the reset model. Due to the stochastic dynamics each probabilistic model displays a different spike output pattern compared to the deterministic LIF neuron. In order to investigate the variation of the neural response of each stochastic model, we repeat the previous scenario 1000 times for each model and compute the corresponding peristimulus time histograms (PSTH). A PSTH is a common neuro-scientific tool that creates a histogram of spikes occurring in a raster plot. A frequency vector is computed which is normalized by dividing each vector element by the number of repetitions and by the size of a time bin (1ms here). Additionally, the raw PSTH is smoothed using Gaussian smoothing with a window with of 10ms. Figure 3 presents a raster plot of the neural response for each stochastic model. The impact of the non-deterministic neural dynamics is clearly visible in the diagrams. We also note that some of the spikes occur in every single repetition resulting in sharp peaks in the PSTH. In this paper, we call these spikes reliable. Reliable spikes are of particular interest in the context of a training algorithm that learns to map a reservoir response to a desired class label. We elaborate on this point as part of the experiments presented below.
S. Schliebs, N. Nuntalid, and N. Kasabov
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Continuous noisy threshold repetition #
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repetition #
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Fig. 3. Raster plot of the neural response of a single stochastic neuron recorded in 1000 repetitions (top row). The bottom row presents the corresponding smoothed PSTH for each raster plot. Each column corresponds to a different stochastic neural model as indicated by the plot titles.
3 Experiments As a proof of concept of the suitability of the proposed probabilistic spatio-temporal filter, we demonstrate here its separation ability. Our experiments are inspired by the study presented in [5] where the authors investigate different neural models in the context of a LSM. It was shown that the choice of a particular neural model impacts the separation ability of the liquid significantly. In our experiments we construct a liquid having a small-world inter-connectivity pattern as described in [9]. A recurrent SNN is generated by aligning 1000 neurons in a three-dimensional grid of size 10 × 10 × 10. Two neurons A and B in this grid are connected with a connection probability P (A, B) = C × e
−d(A,B) λ2
(5)
where d(A, B) denotes the Euclidean distance between two neurons and λ corresponds to the density of connections which was set to λ = 2 in all simulations. Parameter C depends on the type of the neurons. We discriminate into excitatory (ex) and inhibitory (inh) neural types resulting in the following parameters for C: Cex−ex = 0.3, Cex−inh = 0.2, Cinh−ex = 0.4 and Cinh−inh = 0.1. The network contained 80% excitatory and 20% inhibitory neurons. All parameter values are directly adopted from [5]. Four recurrent SNN are generated each employing a different neural model. All networks have the same network topology and the same connection weight matrix. The networks are then stimulated by two input spike trains independently generated by a Poisson process with a mean rate of 200Hz. The response of each network was recorded in 25 independent repetitions. The averaged response of the networks is presented in Figure 4. The two top rows of diagrams show the average raster plots of the spike activity for both stimuli. The
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Fig. 4. Averaged spike response of the reservoirs using different neural models to two independent input stimuli recorded in 25 independent runs (top two rows). The averaged normalized Euclidean differences in each time bin (bin size 1ms) between the responses are presented in the bottom row.
darker the shade in these plots, the more likely the corresponding neuron was observed to spike at the given time bin in 25 runs. The shade is white for time bins in which no neural activity was observed in any run. Each time bin has a size of 1ms. Similar to the raster plots in Figure 3 we notice some reliable spikes in the response corresponding to the very dark shades in the plots. The bottom row of diagrams in Figure 4 presents the averaged normalized Euclidean distances between the two responses for each time bin. The very same distance measure was used in [5] to evaluate differences in response patterns. We note the comparably low separation ability of the deterministic LIF model which confirms the findings in [5]. The results indicate that stochastic models have the potential to significantly increase the separation ability of the reservoir. However, additional experimental analysis is required to provide strong statistical evidence for this claim.
4 Conclusion and Future Directions This study has proposed an extension of the eSNN architecture that enables the method to process spatio-temporal data. The extension projects a spatio-temporal signal into a single high-dimensional network state that can be learned by the eSNN training algorithm. We have proposed a novel probabilistic LSM approach that was subjected to an initial feasibility analysis. It was experimentally demonstrated that probabilistic neural models are principally suitable reservoirs that have furthermore the potential to enhance the separation ability of the system. Future studies will investigate the characteristics of the extended eSNN on typical benchmark problems. Furthermore, the method is intended to be used to classify
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real-world EEG data sets. Particularly interesting are also different stochastic neural models, such as the dynamic stochastic synapses discussed in [10].
References 1. Bohte, S.M., Kok, J.N., Poutr´e, J.A.L.: Error-backpropagation in temporally encoded networks of spiking neurons. Neurocomputing 48(1-4), 17–37 (2002) 2. Clopath, C., Jolivet, R., Rauch, A., L¨uscher, H.R., Gerstner, W.: Predicting neuronal activity with simple models of the threshold type: Adaptive exponential integrate-and-fire model with two compartments. Neurocomput. 70(10-12), 1668–1673 (2007) 3. Destexhe, A., Contreras, D.: Neuronal computations with stochastic network states. Science 314(5796), 85–90 (2006) 4. Gerstner, W., Kistler, W.M.: Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge University Press, Cambridge (2002) 5. Grzyb, B.J., Chinellato, E., Wojcik, G.M., Kaminski, W.A.: Which model to use for the liquid state machine? In: IJCNN 2009: Proceedings of the 2009 international joint conference on Neural Networks, pp. 1692–1698. IEEE Press, Piscataway (2009) 6. van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (2007) 7. Kasabov, N.: The ECOS framework and the ECO learning method for evolving connectionist systems. JACIII 2(6), 195–202 (1998) 8. Kasabov, N.: To spike or not to spike: A probabilistic spiking neuron model. Neural Networks 23(1), 16–19 (2010) 9. Maass, W., Natschl¨ager, T., Markram, H.: Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Computation 14(11), 2531–2560 (2002) 10. Maass, W., Zador, A.: Dynamic stochastic synapses as computational units. In: Advances in Neural Information Processing Systems, pp. 903–917. MIT Press, Cambridge (1998) 11. Schliebs, S., Defoin-Platel, M., Kasabov, N.: Analyzing the dynamics of the simultaneous feature and parameter optimization of an evolving spiking neural network. In: International Joint Conference on Neural Networks, IEEE - INNS - ENNS. IEEE Computer Society Press, Barcelona (2010) 12. Schliebs, S., Defoin-Platel, M., Worner, S., Kasabov, N.: Integrated feature and parameter optimization for an evolving spiking neural network: Exploring heterogeneous probabilistic models. Neural Networks 22(5-6), 623–632 (2009) 13. Schrauwen, B., Verstraeten, D., Campenhout, J.V.: An overview of reservoir computing: theory, applications and implementations. In: Proceedings of the 15th European Symposium on Artificial Neural Networks, pp. 471–482 (2007) 14. Thorpe, S.J.: How can the human visual system process a natural scene in under 150ms? On the role of asynchronous spike propagation. In: ESANN. D-Facto public (1997) 15. Verstraeten, D., Schrauwen, B., D’Haene, M., Stroobandt, D.: An experimental unification of reservoir computing methods. Neural Networks 20(3), 391–403 (2007) 16. Wysoski, S.G., Benuskova, L., Kasabov, N.K.: Adaptive learning procedure for a network of spiking neurons and visual pattern recognition. In: Blanc-Talon, J., Philips, W., Popescu, D., Scheunders, P. (eds.) ACIVS 2006. LNCS, vol. 4179, pp. 1133–1142. Springer, Heidelberg (2006) 17. Yamazaki, T., Tanaka, S.: 2007 special issue: The cerebellum as a liquid state machine. Neural Netw. 20(3), 290–297 (2007)
