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Estimating neural connection strengths from firing intervals

Maren Bråthen Kristoffersen, Bjørn Fredrik Nielsen, Susanne Solem

TL;DR

This paper addresses recovering neural connectivity strengths from firing-interval data in a network described by a Heaviside-activated activity model. By treating firing intervals as known, the nonlinear forward map becomes a set of decoupled linear ODEs, allowing the construction of per-neuron linear systems $\mathsf{A}^{(i)}\mathsf{w}^{(i)}=\mathsf{b}^{(i)}$ that are regularized with TSVD to estimate each row $\mathsf{w}^{(i)}$ of the connectivity matrix $W$. The authors analyze the singular-value structure, propose a practical TSVD-based algorithm, and discuss error propagation and forward-operator continuity under threshold conditions on external inputs. Numerical experiments reveal that the problem is ill-posed, especially for symmetric connectivities, and that regularization is essential to obtain stable reconstructions, with performance improving for non-symmetric networks and larger data sets. The work highlights both the potential and the limitations of extracting quantitative connectivity from firing-interval data in a simplified neuronal model, and outlines future directions for handling unknown initial conditions, time-delay effects, and larger networks.

Abstract

We propose and analyse a procedure for using a standard activity-based neuron network model and firing data to compute the effective connection strengths between neurons in a network. We assume a Heaviside response function, that the external inputs are given and that the initial state of the neural activity is known. The associated forward operator for this problem, which maps given connection strengths to the time intervals of firing, is highly nonlinear. Nevertheless, it turns out that the inverse problem of determining the connection strengths can be solved in a rather transparent manner, only employing standard mathematical tools. In fact, it is sufficient to solve a system of decoupled ODEs, which yields a linear system of algebraic equations for determining the connection strengths. The nature of the inverse problem is investigated by studying some mathematical properties of the aforementioned linear system and by a series of numerical experiments. Finally, under an assumption preventing the effective contribution of the network to each neuron from staying at zero, we prove that the involved forward operator is continuous. Sufficient criteria on the external input ensuring that the needed assumption holds are also provided.

Estimating neural connection strengths from firing intervals

TL;DR

This paper addresses recovering neural connectivity strengths from firing-interval data in a network described by a Heaviside-activated activity model. By treating firing intervals as known, the nonlinear forward map becomes a set of decoupled linear ODEs, allowing the construction of per-neuron linear systems that are regularized with TSVD to estimate each row of the connectivity matrix . The authors analyze the singular-value structure, propose a practical TSVD-based algorithm, and discuss error propagation and forward-operator continuity under threshold conditions on external inputs. Numerical experiments reveal that the problem is ill-posed, especially for symmetric connectivities, and that regularization is essential to obtain stable reconstructions, with performance improving for non-symmetric networks and larger data sets. The work highlights both the potential and the limitations of extracting quantitative connectivity from firing-interval data in a simplified neuronal model, and outlines future directions for handling unknown initial conditions, time-delay effects, and larger networks.

Abstract

We propose and analyse a procedure for using a standard activity-based neuron network model and firing data to compute the effective connection strengths between neurons in a network. We assume a Heaviside response function, that the external inputs are given and that the initial state of the neural activity is known. The associated forward operator for this problem, which maps given connection strengths to the time intervals of firing, is highly nonlinear. Nevertheless, it turns out that the inverse problem of determining the connection strengths can be solved in a rather transparent manner, only employing standard mathematical tools. In fact, it is sufficient to solve a system of decoupled ODEs, which yields a linear system of algebraic equations for determining the connection strengths. The nature of the inverse problem is investigated by studying some mathematical properties of the aforementioned linear system and by a series of numerical experiments. Finally, under an assumption preventing the effective contribution of the network to each neuron from staying at zero, we prove that the involved forward operator is continuous. Sufficient criteria on the external input ensuring that the needed assumption holds are also provided.
Paper Structure (12 sections, 9 theorems, 69 equations, 12 figures, 3 tables)

This paper contains 12 sections, 9 theorems, 69 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Derivation of the method
  4. Singular values
  5. The algorithm
  6. Error considerations in inverse problems
  7. Continuity of the forward operator
  8. Numerical experiments
  9. Mildly or severely ill-posed
  10. Symmetric connectivities
  11. Non-symmetric connectivities
  12. Concluding remarks

Key Result

Lemma 3.1

Assume that $\{ I_i^k \}$ is in the range of the operator $F$. Then, any solution $\mathsf{W}=[W_{ij}]$ of must satisfy, for any $i \in \Gamma$ and for any $t \in \left\{t_i^k, \, t_i^k+\Delta t_i^k\right\}_{k=1}^{K(i)}$, the linear equation whenever $B_i$ is continuous at $t$, and the inequalities where $s_j$ solves eq:motherODE_fire for $j=1, \, 2, \, \ldots, \, n$.

Figures (12)

  • Figure 1: Illustration of Remark \ref{['remark:beta']}. When $f$ (in blue) satisfies $|f(\tilde{t})| >\gamma^{-1}$ for a time $\tilde{t}$, i.e., the point $(\tilde{t}, f(\tilde{t}))$ is outside the green area, any function $g(t)$ satisfying $|f(t)-g(t)|<\gamma^{-1}$, i.e., the graph of $g(t)$ is within the red area, will have the same sign as $f(t)$ at $t=\tilde{t}$.
  • Figure 2: Samples of the true connectivity function for $n=100$.
  • Figure 3: Log-plots of the singular values of $\mathsf{A}^{(i)}$ and their approximations $C m ^{-\alpha}$ and $C e^{-\alpha m}$ for neuron $i=3$. The left and right plots show these quantities for the symmetric and non-symmetric connectivities, respectively.
  • Figure 4: Results for $n=20$ neurons with noise added to $\mathsf{b}^{(i)}$ and where the true connectivity is symmetric. The upper row shows the inverse solution $\mathsf{W}_{\text{inv}}^\kappa$ computed with various degree of noise. The lower row displays $\mathsf{W}_{\text{inv}}^\kappa-\mathsf{W}$.
  • Figure 5: The inverse solution for $n=100$ neurons with noise added to $\mathsf{b}^{(i)}$ with a $1\%$ noise level and where the true connectivity is symmetric. The minimal Frobenius norm is used to calculate the regularization parameter, and the regularization parameter is $\kappa=22$.
  • ...and 7 more figures

Theorems & Definitions (24)

  • Definition 2.1: Firing
  • Definition 2.2: Symmetric difference halmos1960naive
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Corollary 3.3
  • Remark 3.4
  • Theorem 3.5
  • proof
  • ...and 14 more