TRAKR – A reservoir-based tool for fast and accurate classification of neural time-series patterns

1 INTRODUCTION . The size and complexity of neural data collected has increased greatly ( Marblestone et al . ( 2013 ) ) . Neural data display rich dynamics in the firing patterns of neurons across time , resulting from the recurrently connected circuitry in the brain . As our insight into these dynamics increases through new recording modalities , so does the desire to understand how dynamical patterns change across time and , ultimately , give rise to different behaviors . A lot of work in computational neuroscience over the past decade has focused on modeling the collective dynamics of a population of neurons in order to gain insight into how firing patterns are related to task variables ( Márton et al . ( 2020 ) ; Richards et al . ( 2019 ) ; Yang et al . ( 2018 ) ; Remington et al . ( 2018 ) ; Kell et al . ( 2018 ) ; Zeng et al . ( 2018 ) ; Pandarinath et al . ( 2018 ) ; Durstewitz ( 2017 ) ; Chaisangmongkon et al . ( 2017 ) ; Rajan et al . ( 2016 ) ; Sussillo et al . ( 2015 ) ; Mante et al . ( 2013 ) ; Sussillo & Barak ( 2013 ) ; Barak et al . ( 2013 ) ; Sussillo & Abbott ( 2009 ) ) . These approaches , however , rely on fitting the whole dynamical system through many rounds of optimization , either indirectly by modeling the task inputs and outputs ( Márton et al . ( 2020 ) ; Kell et al . ( 2018 ) ; Chaisangmongkon et al . ( 2017 ) ; Sussillo et al . ( 2015 ) ; Mante et al . ( 2013 ) ; Sussillo & Barak ( 2013 ) , or directly by fitting the weights of a neural network to recorded firing patterns ( Pandarinath et al . ( 2018 ) ; Durstewitz ( 2017 ) ) . Thus , these approaches can be too time- and computation-intensive for certain applications , e.g . in clinical settings where decisions need to be taken based on recordings in real-time . In these settings , neural time-series patterns need to be accurately distinguished in order to , say , detect the onset of seizures , or distinguish different mental states . Previous approaches to classifying time series lie on a spectrum from simple distance metrics ( e.g. , Euclidean ) to more computationally intensive approaches such as dynamic time warping ( Xing et al . ( 2010 ) ) , ensembles of classifiers ( Bagnall et al . ) or deep supervised learning ( Jeong ( 2020 ) ; Fawaz et al . ( 2019 ) ) . Computing simple distance metrics is fast and straightforward , but does not always yield high accuracy results because the patterns may not be perfectly aligned in time . On the other end of the spectrum , ensembles of classifiers and deep learning-based approaches ( Bagnall et al . ; Jeong ( 2020 ) ; Fawaz et al . ( 2019 ) ) have been developed that can offer high accuracy results , but at high computational cost . Dynamic time warping ( DTW ) has been consistently found to offer good results in practice relative to computational cost ( Fawaz et al . ( 2019 ) ; Bagnall et al . ( 2016 ) ; Serrà & Arcos ( 2014 ) ) and is currently routinely used to measure the similarity of time-series patterns . Previous work in reservoir computing has shown that networks of neurons can be used as reservoirs of useful dynamics , so called echo-state networks ( ESNs ) , without the need to train recurrent weights through successive rounds of expensive optimization ( Vlachas et al . ( 2020 ) ; Pathak et al . ( 2018 ) ; Vincent-Lamarre et al . ( 2016 ) ; Buonomano & Maass ( 2009 ) ; Jaeger & Haas ( 2004 ) ; Jaeger ( a ; b ) ; Maass et al . ( 2002 ) ) . This suggests reservoir networks could offer a computationally cheaper alternative to deep supervised approaches in the classification of neural time-series data . However , the training of reservoir networks has been found to be more unstable compared to methods that also adjust the recurrent connections ( e.g. , via backpropagation through time , BPTT ) in the case of reduced-order data ( Vlachas et al . ( 2020 ) ) . Even though ESNs have been shown to yield good results when fine-tuned ( Tanisaro & Heidemann ( 2016 ) ; Aswolinskiy et al . ( 2016 ) ) , convergence represents a significant problem when training ESNs end-to-end to perform classification on complex time-series datasets , and is a hurdle to their wider adoption . Here , we propose fitting the reservoir output weights to a single time series - thus avoiding many rounds of training that increase training time and could potentially cause instabilities . We use the error generated through the output unit in response to a particular test pattern as input to a classifier . We show that using this approach , we obtain high accuracy results on a benchmark dataset - sequential MNIST - outperforming other approaches such as simple distance metrics ( e.g. , based on Euclidean distance or Mutual Information ) and a previous approach based on echo-state networks , while performing on par with deep supervised networks . We also show that our approach is more robust to noise than other approaches , in particular deep supervised networks . At the same time , TRAKR achieves high performance while keeping training and inference time low . We also apply our tool , TRAKR , to neural data from the macaque orbitofrontal cortex ( OFC ) , a higher-order brain region involved in encoding expectations , and inducing changes in behavior during unexpected outcomes ( Rich & Wallis ( 2016 ) ; Rudebeck & Rich ( 2018 ) ; Jones et al . ( 2012 ) ; Wallis ( 2012 ) ; Schoenbaum ( 2009 ) ; Burke et al . ( 2009 ) ; Wallis & Miller ( 2003 ) ; Schoenbaum et al . ( 1998 ) ) . This data consists of 128-channel electrocorticography ( micro-ECOG ) recordings obtained from the macaque OFC , including anterior and posterior areas 11 and 13 , during a reward expectation task . The task was designed to understand if and how expectations encoded in OFC are updated by unexpected outcomes . TRAKR is able to distinguish three different behaviorally relevant epochs based on the neural time-series with high accuracy , revealing that OFC units differentiate between different task episodes . This shows it can be used as a reliable tool to gain further insight into the information encoded in neural circuits . Taken together , we make the following contributions : - We show that by fitting single time series and working with the error signal reservoirs can perform on par with supervised deep networks in time series classification - We show that this approach , while easy to use , outperforms other commonly used approaches , such as Dynamic Time Warping ( DTW ) , and other distance measures embraced for their simplicity - We show that our approach is more robust to noise than other approaches - Our approach is computationally less expensive than other approaches , achieving high performance at low train- and inference- time - It is able to detect differences between neural patterns with high accuracy Altogether , this shows that TRAKR is a viable alternative to state-of-the-art approaches for time series classification . 2 METHODS . 2.1 MODEL DETAILS . TRAKR ( Figure 1A ) is a reservoir-based recurrent neural network ( RNN ) with N recurrently connected neurons . Recurrent weights , J , are initialized randomly and remain aplastic over time ( Buonomano & Maass ( 2009 ) ; Jaeger ( b ) ; Maass et al . ( 2002 ) ) . The readout unit , zout , is connected to the reservoir through a set of output weights , wout , which are plastic and are adjusted during training . The reservoir also receives an input signal , I ( t ) , through an aplastic set of weights win . The network is governed by the following equations : τ dxi ( t ) dt = −xi ( t ) + g N∑ j=1 Jijϕj ( t ) + wi , inI ( t ) ( 1 ) zout ( t ) = ∑ i wout , i ( t ) xi ( t ) ( 2 ) Here , xi ( t ) is the activity of a single neuron in the reservoir , τ is the integration time constant , g is the gain setting the scale for the recurrent weights , and J is the recurrent weight matrix of the reservoir . The term g N∑ j=1 Jijϕj ( t ) denotes the strength of input to a particular neuron from other neurons in the reservoir and I ( t ) is the input signal ( Equation 1 ) . zout ( t ) denotes the activity of the readout unit together with the output weights , wout ( Equation 2 ) . In our notation , wij denote the weight from neuron j to i , and so wout , i means the weight from ith unit in the reservoir to the readout unit . ϕ is the activation function given by : ϕi ( t ) = tanh ( xi ( t ) ) ( 3 ) We use recursive least squares ( RLS ) to adjust the output weights , wout during training ( Haykin , Simon S. ( 1996 ) ) . The algorithm and the update rules are given by : ∆wout , i ( t ) = −η ( zout ( t ) − f ( t ) ) ∑ j Pij ( t ) ϕj ( t ) ( 4 ) wout , i ( t ) = wout , i ( t− 1 ) + ∆wout , i ( t ) ( 5 ) Here , η is the learning rate , f ( t ) is the target function , and the term ∑ j Pij ( t ) ϕj ( t ) acts as a regularizer where P is the inverse cross-correlation matrix of the network firing rates . For details on setting hyperparameters , see Appendix A . 2.2 ADJUSTING RESERVOIR DYNAMICS . During training , the output weights , wout , are optimized using RLS based on the instantaneous difference between the output , zout ( t ) , and the target function , f ( t ) . This optimization is performed in one shot ( without the need for multiple optimization rounds ) . Here , we use the reservoir to autoencode the input signal , thus f ( t ) = I ( t ) . The instantaneous difference gives rise to an error term , E ( t ) , calculated as : E ( t ) = N∑ i=1 ∆wout , i ( t ) ( 6 ) 2.3 OBTAINING THE ERROR SIGNAL . After training , the output weights , wout are frozen . The test pattern is fed to the network via the input , I ( t ) , and the network is iterated to obtain the error , E ( t ) over the duration of the test signal . The error , E ( t ) is computed as the difference between the test signal and the network output ( Equation 6 ) . The error may vary depending on the similarity of a given test signal to the learned time series . The error is used as input to a classifier .

Summary: Here, the authors propose a reservoir-computing based framework for time-series applications. The proposed methodology involves training the output units of the echo state network to recapitulate, or autoencode, a test signal. Next, time-series are fed to the network and the error signal between the time-series and the network's response is used as input to an SVM that is used for classification.  Effectively the error signal is used as a distance metric.  The authors report that the network performs well on sequential MNIST and decoding of neural data. Additionally, they argue that a key advantage of their approach is that it is computationally lightweight – both training and inference are fast.