Traversing the region between these examples by progressively decreasing the synaptic threshold, we found a compensatory increase in the reliance GDC-0941 cost on higher recruitment threshold neurons and decrease in reliance upon lower recruitment threshold neurons, especially for inhibition (Figures 4G and S5). This path through parameter space represents an insensitive direction of movement along the model cost-function surface, with a tradeoff between the use of synaptic and recruitment thresholds. We next asked which features of the circuit connectivity were necessary and which could be changed with minimal degradation of model performance. To address this question, we performed a sensitivity analysis on the connections

weights for circuits based on both the synaptic threshold and neuronal recruitment-threshold mechanisms. For a given form of the synaptic activation function, we first determined the best-fit connectivity pattern from the minimum of our fit cost function. We then asked how the cost function changed when individual synaptic connections were altered from their best-fit values, and which concerted patterns of synaptic connection changes caused the greatest changes in the fit performance. These quantities were found by calculating, for each neuron, how rapidly the cost function curved away from its minimum value when the presynaptic Docetaxel weights onto the neuron were varied around their best-fit values.

Mathematically, this curvature is defined by the sensitivity (or Hessian) matrix Hij(k) whose (i,j)th(i,j)th element contains the second derivative of the cost function with respect to changes in the weights of the ith and jth presynaptic inputs

onto neuron k ( Figure 6A). Sensitivity to changes in a single presynaptic input weight are given by the diagonal elements of the matrix. Sensitivity to Cell press concerted patterns of weight changes are found from the eigenvector decomposition of the matrix. Eigenvectors corresponding to the largest eigenvalues give the patterns of weight changes along which the cost function curves most sharply, and hence identify the most sensitive directions of the circuit to perturbations. Eigenvectors corresponding to small eigenvalues define patterns of weight changes to which the cost function is insensitive. Figure 6A shows the sensitivity matrix for a neuron from the synaptic threshold model of Figure 4C. The sensitivity matrix separates into diagonal blocks, indicating that changes in the cost function due to perturbations in excitatory (inputs 1–25) and inhibitory (26–50) weights were nearly independent of one another. Within these blocks, the precise grid-like pattern of sensitivities was dependent upon the exact choice of tuning curves used in any given simulation and was removed by averaging the sensitivity matrices of 100 circuit simulations with different random draws of tuning curves (Figure 6B).