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Connectomic constraints on computation in feedforward networks of spiking neurons

Abstract

Several efforts are currently underway to decipher the connectome or parts thereof in a variety of organisms. Ascertaining the detailed physiological properties of all the neurons in these connectomes, however, is out of the scope of such projects. It is therefore unclear to what extent knowledge of the connectome alone will advance a mechanistic understanding of computation occurring in these neural circuits, especially when the high-level function of the said circuit is unknown. We consider, here, the question of how the wiring diagram of neurons imposes constraints on what neural circuits can compute, when we cannot assume detailed information on the physiological response properties of the neurons. We call such constraints—that arise by virtue of the connectome—connectomic constraints on computation. For feedforward networks equipped with neurons that obey a deterministic spiking neuron model which satisfies a small number of properties, we ask if just by knowing the architecture of a network, we can rule out computations that it could be doing, no matter what response properties each of its neurons may have. We show results of this form, for certain classes of network architectures. On the other hand, we also prove that with the limited set of properties assumed for our model neurons, there are fundamental limits to the constraints imposed by network structure. Thus, our theory suggests that while connectomic constraints might restrict the computational ability of certain classes of network architectures, we may require more elaborate information on the properties of neurons in the network, before we can discern such results for other classes of networks.

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Notes

  1. 1.

    1 We assume a single fixed absolute refractory period for all neurons, for convenience, although our results would be no different if different neurons had different absolute refractory periods.

  2. 2.

    2 Models such as the Leaky Integrate-and-Fire (LIF) and Spike Response Model (SRM), in addition to the constraints in our model have their membrane potential function P(·) specified outright. In case of the LIF model, this is specified via a differential equation and in the case of SRM, the specific functional form is written down explicitly.

  3. 3.

    In this work, we do not treat electrical synapses or ephaptic interactions (Shepherd 2004).

  4. 4.

    We do not treat stochastic variability in the responses of neurons or neuromodulation in this paper.

  5. 5.

    In many biological neurons, the membrane potential that the soma (or axon initial segment) must reach, in order to elicit a spike is not fixed at all times and is, for example, a function of the inactivation levels of the voltage-gated Sodium channels. Our model can accomodate this phenomenon, to the extent that this threshold itself is a function of spikes afferent in the past ϒ milliseconds and spikes efferent from the present neuron in the past ρ milliseconds.

  6. 6.

    This is violated, notably, in neurons that have a post-inhibitory rebound.

  7. 7.

    The interested reader is referred to Online Resource B for a discussion on the issue of infinitely-long input spike-trains in this context.

  8. 8.

    Note that for sufficiently small values of T (in relation to Υ and ρ), no χ may satisfy a T-Gap Criterion. This is deliberate formulation that will minimize notational clutter in forthcoming definitions.

  9. 9.

    The classes of networks could correspond to ones that contain all networks with specific network architectures, although for the purpose of the definition, there is no reason to require this to be the case.

  10. 10.

    Strictly speaking, the output spike happens at 4i+3+𝜖, where 𝜖>0 is a small real number. Henceforth whenever we say an output spike is after a certain time instant, we mean it in this sense.

  11. 11.

    Equipped with instances of our model neurons

  12. 12.

    As a by-product, the proof also ends up providing a complete characterization of the set of transformations spanned by the set of all feedforward networks equipped with neurons of the present abstract model, which turns out to be exactly this “nice” set.

  13. 13.

    Recall that when we say transformation, without further qualification, we mean one, of the form \(\mathcal {T}:\mathcal {F}_{m}\rightarrow \mathcal {S}\).

  14. 14.

    Strictly speaking, it turns out that this is not true; axiom 2 may be violated.

  15. 15.

    Which we call a “reset gap” from now on, for the sake of exposition.

  16. 16.

    Whenever we say decimal expansion, we forbid decimal expansions with an infinite number of successive 9s. With this restriction, each real number has a unique decimal expansion.

  17. 17.

    This in itself is a formidable problem and one that is taking heroic effort.

  18. 18.

    See Fig.  7a and the second paragraph of Section 8.

  19. 19.

    This characterization is a consequence of Theorem 4. In particular, it is the set of all transformations that are not causal, time-invariant or resettable.

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Acknowledgments

This work was supported, in part, by a National Science Foundation grant (NSF IIS-0902230) to A.B.

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Correspondence to Venkatakrishnan Ramaswamy.

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Conflict of interest

The authors declare that they have no conflict of interest.

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Ramaswamy, V., Banerjee, A. Connectomic constraints on computation in feedforward networks of spiking neurons. J Comput Neurosci 37, 209–228 (2014) doi:10.1007/s10827-014-0497-5

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Keywords

  • Spiking neurons
  • Connectomics
  • Feedforward networks