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Causal pattern recovery from neural spike train data using the Snap Shot Score

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Abstract

We present a new approach to learning directed information flow networks from multi-channel spike train data. A novel scoring function, the Snap Shot Score, is used to assess potential networks with respect to their quality of causal explanation for the data. Additionally, we suggest a generic concept of plausibility in order to assess network learning techniques under partial observability conditions. Examples demonstrate the assessment of networks with the Snap Shot Score, and neural network simulations show its performance in complex situations with partial observability. We discuss the application of the new score to real data and indicate how it can be modified to suit other neural data types.

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Notes

  1. Other choices for the decay constant are possible: the smaller d is chosen, the larger the range of time-lags considered for detection of interrelations. Extreme values where d = 0 (activity level constantly 1 once a spike occurred on the channel) or d ≈ 0 (activity decaying extremely slow) are unlikely to deliver sensible results. We have chosen d = 1/3 to keep examples (Section 3) expressive and clear. For real data, the decay constant can be derived from the anticipated maximal causal lag (in time-bins) or by using a parameter series as outlined in Example 3, later.

  2. A loop-link renders a node parent and child at the same time. We will refer to such configurations as self-exciting.

  3. Implementation note: Profiling implementations (in C (Kernighan and Ritchie 1988) and Python (van Rossum et al. 2009)) of our method revealed that calculating joins is computationally much more expensive than calculating the SSS value. Instead of recalculating a join for different nodes, we suggest to perform scoring join-wise, i.e. scoring all nodes with a join once it has been computed.

  4. Generally the mean score is unknown, because an exhaustive evaluation is computationally impossible in practical dimensions.

  5. For independent random spike trains, SSS values of all parent configurations lie within mean ± SD and approach mean score (\(SD \searrow 0\)) for increasing length of spike trains (not shown).

  6. Start- and end-nodes underlined, nodes on path in full network (given for illustration) italic, hidden nodes in brackets.

  7. A series of directed links is called a path, if the origin of all links equals their predecessors’ destination. Length l(a →b) of directed path from node a to b is defined as number of links on path. The length of a path a →b directly corresponds to the time-lag a signal needs to propagate from a to b. In our neural simulation, the time-lag in time-bins (1 ms) is equal to the length of a path.

  8. For l min  = 1 there exist 12 (≈ 6.6% out of 182 possible links, l max  = 1), 27 (≈ 14.8%, l max  = 2), 34 (≈ 18.7%, l max  = 3), 37 (≈ 20.0%, l max  = 4) plausible links.

  9. Note that the definition of the recovery rate corresponds to sensitivity. We have called it differently because we do not expect the recovery rate to reach 100% (as is explained in the text later), which the reader might assume if confronted with the familiar but misleading term sensitivity.

  10. The chance level for at least h hits out of p plausible links out of N total links with k links learned is \( \sum_{i=h}^{\min\{p,k\}}q_i\), where \(q_i = \frac{\binom{p}{i}\binom{N-p}{k-i}}{\binom{N}{k}}\).

  11. The \(\textit{impetus}=100\cdot\#\textit{evoked spikes}/{\#\textit{stimulating spikes}}\) where number of \(\textit{evoked spikes} = \#\textit{spikes in simulation output} - \#\textit{stimulating spikes}\). If, for example, impetus=0%, then no spikes were evoked by the stimulation and the simulation output is equal to uncorrelated random spike trains. For impetus=100%, the simulation output is a mixture of two halves: stimulation spikes and evoked spikes.

  12. The data sets sum up to a total recording time of 5 min. The impetus was found between 29.7% and 30.0%. For an impetus of 30%, the simulation output consists of 3 evoked spikes per 10 uncorrelated stimulation spikes (on average).

  13. Activity level series correspond to rate-limited firing rate series that were computed using a causal kernel. A causal kernel function f satisfies \(\text{supp}(f) \cap \mathbb{R}_{<0} = \{ x \in \mathbb{R} | f(x)>0 \} \cap \mathbb{R}_{<0} = \emptyset\) (Dayan and Abbott 2005, p. 14). Such a kernel does not make use of information gained in the future. The activity level of channel k at time t is given by a k,t = ∑ j = 0,...,t f( t − j ·s k,t − j ) .

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Acknowledgements

We thank two anonymous reviewers for a thorough review and helpful comments. This work was supported by the CARMEN e-science project (www.carmen.org.uk) funded by the EPSRC (EP/E002331/1).

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  1. School of Biology, University of St Andrews, St Andrews, KY16 9TS, UK

    Christoph Echtermeyer & V. Anne Smith

  2. Institute of Neuroscience, The Henry Wellcome Building for Neuroecology, Newcastle University, Newcastle upon Tyne, NE2 4HH, UK

    Tom V. Smulders

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  1. Christoph Echtermeyer
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Echtermeyer, C., Smulders, T.V. & Smith, V.A. Causal pattern recovery from neural spike train data using the Snap Shot Score. J Comput Neurosci 29, 231–252 (2010). https://doi.org/10.1007/s10827-009-0174-2

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