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Reconstruction of sparse recurrent connectivity and inputs from the nonlinear dynamics of neuronal networks

Abstract

Reconstructing the recurrent structural connectivity of neuronal networks is a challenge crucial to address in characterizing neuronal computations. While directly measuring the detailed connectivity structure is generally prohibitive for large networks, we develop a novel framework for reverse-engineering large-scale recurrent network connectivity matrices from neuronal dynamics by utilizing the widespread sparsity of neuronal connections. We derive a linear input-output mapping that underlies the irregular dynamics of a model network composed of both excitatory and inhibitory integrate-and-fire neurons with pulse coupling, thereby relating network inputs to evoked neuronal activity. Using this embedded mapping and experimentally feasible measurements of the firing rate as well as voltage dynamics in response to a relatively small ensemble of random input stimuli, we efficiently reconstruct the recurrent network connectivity via compressive sensing techniques. Through analogous analysis, we then recover high dimensional natural stimuli from evoked neuronal network dynamics over a short time horizon. This work provides a generalizable methodology for rapidly recovering sparse neuronal network data and underlines the natural role of sparsity in facilitating the efficient encoding of network data in neuronal dynamics.

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Acknowledgements

This work was supported by NSF grant DMS-1812478 and a Swarthmore Faculty Research Support Grant.

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Appendix: Compressive sensing theory

Appendix: Compressive sensing theory

Compressive sensing (CS) theory demonstrates that for signals with a sparse representation, the number of dominant components in the sparse domain determines the minimum sampling rate necessary for an accurate reconstruction. Improving upon conventional signal acquisition theory, which generally asserts that the sampling rate should instead be determined by the total number of signal components (Shannon, 1949), CS theory provides an important new direction for efficient signal sampling and subsequent reconstruction (Donoho, 2006; Candes et al., 2006). Considering that common signals and sensory stimuli, such as scenes, soundwaves, and odorants, as well as network connectivity are sparse in an appropriate domain (Field, 1994; Markram et al., 1997; Ganmor et al., 2011; He et al., 2007), CS theory has amassed numerous and broad scientific applications (Gross et al., 2010; Lustig et al., 2007; Dai et al., 2009; Berger et al., 2010).

In developing the mathematical framework for CS theory, we consider recovering an n-component signal, \(\mathbf {x}\), using a set of weighted linear measurements. Assuming m weighted measurements are utilized, the sampling scheme takes the form of an \(m \times n\) sampling matrix, \(\mathbf {A}\), such that each row contains a single weighted measurement. Reconstructing the signal \(\mathbf {x}\) thus requires solving linear system \(\mathbf {Ax}=\mathbf {b}\), where \(\mathbf {b}\) is the m-vector obtained from sampling.

To reconstruct the sparsest, and thus most compressible, solution, we must select the \(\mathbf {x}\) with the minimal number of non-zero components that satisfies \(\mathbf {Ax}=\mathbf {b}\). Since this problem cannot be solved in polynomial time (Bruckstein et al., 2009), CS theory demonstrates that minimizing \(\vert \mathbf {x} \vert _{ \ell _1} = \displaystyle \sum _{i=1}^n |x_i|\) yields a reconstruction equivalent to finding the sparsest \(\mathbf {x}\) for a large class of sampling matrices (Candes and Wakin, 2008). The reconstruction thus requires solving

$$\begin{aligned} \arg \min _{\mathbf {x}\in R ^n} \vert \mathbf {x} \vert _{ \ell _1} \text{ subject } \text{ to } \mathbf {Ax}=\mathbf {b}. \end{aligned}$$
(15)

This specific \(\ell _1\) minimization problem can be efficiently solved using a host of algorithms, such as the orthogonal matching pursuit (OMP), the least angle regression (LARS), and the least absolute shrinkage and selection operator (LASSO) methods (Tropp and Gilbert, 2007; Donoho and Tsaig, 2008), making the reconstruction computationally feasible even for relatively large signals. Moreover, if signal \(\mathbf {x}\) is not sparse in the sampled domain, but is instead sparse under a transform, L, then the linear system \(\varvec{\phi } \mathbf {\hat{x}} =\mathbf {b},\) where \(\varvec{\phi }=\mathbf {A}L^{-1}\) and \(\mathbf {\hat{x}} = L\mathbf {x}\), can be considered similarly. In this particular work, OMP is applied in all reconstructions. The recovered recurrent connectivity exhibits minor variations when utilizing other standard optimization algorithms, but the optimal reconstruction accuracy is comparable in each case.

In determining an appropriate sampling scheme, it is important to note CS theory demonstrates that sampling matrices exhibiting sufficiently little correlation among their columns and approximately preserving the magnitude of sampled signals generally yield successful reconstructions of sparse signals (Candes and Wakin, 2008; Baraniuk, 2007). A broad class of matrices demonstrating randomness in their structure have been proven to exhibit these properties (Candes et al., 2006; Candes and Wakin, 2008), and there are consequently numerous sampling schemes amenable to CS reconstruction.

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Barranca, V.J. Reconstruction of sparse recurrent connectivity and inputs from the nonlinear dynamics of neuronal networks. J Comput Neurosci (2022). https://doi.org/10.1007/s10827-022-00831-x

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  • DOI: https://doi.org/10.1007/s10827-022-00831-x

Keywords

  • Network reconstruction
  • Nonlinear dynamics
  • Mean-field analysis
  • Signal processing
  • Integrate-and-fire model networks