Flexible neural connectivity under constraints on total connection strength

Neural computation is determined by neurons’ dynamics and circuit connectivity. Uncertain and dynamic environments may require neural hardware to adapt to different computational tasks, each requiring different connectivity configurations. At the same time, connectivity is subject to a variety of constraints, placing limits on the possible computations a given neural circuit can perform. Here we examine the hypothesis that the organization of neural circuitry favors computational flexibility: that it makes many computational solutions available, given physiological constraints. From this hypothesis, we develop models of connectivity degree distributions based on constraints on a neuron’s total synaptic weight. To test these models, we examine reconstructions of the mushroom bodies from the first instar larva and adult Drosophila melanogaster. We perform a Bayesian model comparison for two constraint models and a random wiring null model. Overall, we find that flexibility under a homeostatically fixed total synaptic weight describes Kenyon cell connectivity better than other models, suggesting a principle shaping the apparently random structure of Kenyon cell wiring. Furthermore, we find evidence that larval Kenyon cells are more flexible earlier in development, suggesting a mechanism whereby neural circuits begin as flexible systems that develop into specialized computational circuits.

. Those bounds are parameterized by the variance of that gaussian, σ 2 α . For each cell type, we varied σ 2 α between its lower and upper bounds and computed the posterior odds at each σ 2 α . Since those bounds differ between cell types, we normalized them to compare the odds ratios. (b) Same as a), for the simplex area vs the zero-truncated binomial model. (c) Posterior odds ratio for the simplex volume vs the zero-truncated binomial model. (d-f) Same as (a-c), but with the Poisson Jeffreys prior for α, p(α) ∝ 1/ √ σ.      c d

Materials and methods
Here we reproduce the Methods sections referenced in the supporting figure captions; the content is the same as in the corresponding main text.

Model comparison
Under equal prior likelihoods for two models X and Y , the posterior likelihood ratio between two models, X and Y is where i indexes data points. We consider the Laplace approximations for the posterior odds, obtained by writing p X = exp ln p X and Taylor expanding the log likelihood ln p X in α around its maximum likelihood value, Truncating at second order then yields a tractable Gaussian integral over the unknown parameter: where the integrals run over the allowed range for α and Under a flat prior for non-negative α, the marginal likelihood is: where 1/σ 2 = i 1/σ 2 i . The simplex volume distribution is a truncated Poisson; we might reasonably use the Jeffreys prior for the Poisson distribution, p(α) ∝ 1/ √ α. In that case, the marginal likelihood is where I a is the modified Bessel function of the first kind. We will drop the indices on K,S in most of the remaining sections, reintroducing them where necessary.

Model comparison: bounded net weight model
Under a bounded net weight, the degree distribution is: July 24, 2020 6/12 The normalization constant Z is so the simplex volume distribution is a zero-truncated Poisson distribution. We will make a Laplace approximation for the simplex volume distribution aroundα, leading to the posterior odds Eq 5 (for a flat prior on non-negative α) or Eq 6 (for the Poisson Jeffreys prior). To calculate the Laplace approximation for the posterior odds we needα and σ 2 . The derivatives of ln p V can be calculated directly (again dropping indices over measurements), So we have and the maximum likelihood solution for α satisfies

Model comparison: fixed net weight model
Under the fixed net synaptic weight, our model is that the degree distribution is proportional to the surface area of the simplex: To calculateα and σ 2 we need the derivatives of ln p A . where and we use the identity We next bound ∂ ∂α Z.
July 24, 2020 7/12 For K ≥ 1, 1 + 1/K is bounded above by √ 2 and below by 1. So, Inserting these into the critical point equation forα provides the bounds: We will also need the curvature of ln p A w.r.t. α atα: Similarly to the first derivative, We use the identity The curvature of Z is The final term 1 + 2 K is bounded above by √ 3 and below by 1, so Defining upper and lower bounds for ∂ 2 ∂α 2 ln Z using the upper and lower bounds of the first and second terms in Eq 23 yields: The upper bound for ∂ 2 ∂α 2 ln Z provides an upper bound for σ 2 , while neglecting Z provides a lower bound for σ 2 (since Z ≥ 1 from Eq 13, so that ln Z ≥ 0): The posterior odds for the simplex area are: July 24, 2020 8/12 where σ 2 = 1/ i 1/σ 2 i . We use the upper and lower bounds for σ 2 i to define upper and lower bounds, respectively, for the likelihood's variance: We computeα numerically by maximizing the likelihood, and compute p A (K i |S i ,α) also numerically, estimating Z by ranging over K = 1 to 2 max iSi .
Bounds for the posterior odds of the fixed net weight model The derivative of the posterior odds under the flat prior, Eq 5, with respect to σ is proportional to Since α > 0 and σ > 0, the last term is bounded between 0 and 1. The middle term is proportional to the form x exp(−x 2 /2), which is maximized by 1/ √ e at x = 1. Since 2/πe < 1, the middle term is less than 1 and the derivative of the posterior odds under a flat prior for α, with respect to σ, is non-negative. The upper bound for σ 2 thus provides an upper bound on the posterior odds. We see that the posterior likelihood i p A (K i |S i ) increases from σ 2 L to σ 2 U (reflected in the log posterior odds ratio for the simplex volume vs the simplex area, S1 Figs 1, 2, 4-9).
The derivative of the posterior odds under the Poisson Jeffreys prior, Eq 6, with respect to σ 2 , is proportional to We saw that the posterior odds for the simplex area distribution also increased with σ 2 for the Poisson Jeffreys prior.

Model comparison: zero-truncated binomial model
The marginal likelihood for the zero-truncated binomial with distribution p B is . For connections to larval KCs, we used the total number of traced projection neurons (PNs) and KCs as the binomial parameter N , averaged over the two sides of the brain [1]. For projections from larval KCs, we used the total number of KCs and output neurons, averaged over the two sides, as N . For projection to adult KCs, we used the number of Kenyon cells plus 150 (the estimated number of olfactory PNs) as N [2]. For projections from adult KCs, we used the number of KCs and output neurons labelled in the data as N .
The variance with respect to q is determined as in Eq (4). The derivates of ln p B are, again dropping indices on K, The maximum likelihood parameterq for the zero-truncated binomial, with M samples of K, each with N trials, obeys:q and the variance atq is

Distances in synaptic configuration space
Above we assumed that synaptic weight configurations could travel between different points in the synaptic weight space along straight lines, endowing the K-dimensional synaptic weight space with a Euclidean (or 2-) norm. This amounts to assuming that synaptic weights can vary together. This could be interpreted, for example, as allowing a unit of synaptic weight (a receptor, perhaps) to be transferred directly between connections. An alternative is to assume that synaptic weights must move separately, which corresponds endowing the synaptic weight space with the 1-norm. In the above interpretation this would mean separating the removal of a receptor from one synapse from the addition of a receptor to another synapse. This changes the surface area of the simplex, since its inner radius isJ/K rather thanJ/ √ K: Changing the norm for the synaptic weights leaves the above calculation of the posterior odds for the fixed net weight model mostly unchanged. The factors of √ K in the normalization constant are replaced by K; this removes the square roots in the derivation of the upper bound for the variance with respect to α so that The optimal number of connections can be calculated in the same manner as previously.
The derivative of A 1 with respect to K is (to order 1/K): (38) At a critical point in K, truncating O(1/K 2 ) and higher-order terms yields