Here, we address the evolution of bow-tie architectures using simulations of multilayered systems evolving to fulfill a given input-output goal.

Mathematically speaking, bow-ties evolve when the rank of the input-output matrix describing the evolutionary goal is deficient.

Deficient rank means that the input-output transformation maps inputs to a limited subspace of outputs, of dimension r. Below, we discuss the implications of this concept also for nonlinear systems.

Bow-tie can evolve for a nonlinear input-output relation too, if the input can be more compactly represented with no effect on the output.

Here we considered the input-output relation as the sole force guiding the evolution of the network, however there may be other constraints or processes affecting network structures.

The intermediate layer is called the “waist” [3], “knot” [1] or “core” [4] of the bow-tie and in gene-regulatory networks the ‘input-output’ [5] or ‘selector’ gene [6].

Many developmental gene regulatory networks have bow-tie structures in which a single intermediate gene ( ‘input-output’ or ‘selector’ gene) combines information from multiple patterning genes (the input layers) and then initiates a self-contained developmental program by regulating an array of output genes [5,6] that can produce a large variety of morphologies [17—20].

In developmental gene regulatory networks, modulated expression of the ‘waist’ ( ‘input-output’ or ‘selector’) gene can result in markedly different phenotypes.

Simulations of multilayered network models evolving towards input-output goals

In the linear model, the total input-output relationship of the network is given by the product of the matrices A1, A2,.

Employing this formalism, we evolve these networks to match a desired goal—given by a matrix G. The goal matrix describes the desired output for any possible vector of inputs, and thus defines the entire input-output function.

The composite model outperforms both the multiplicative and additive models, though less so when Ca2+ conductance is decreased by 75%, suggesting that the inability of the multiplicative and additive models to represent the input-output relationship depends on dendritic electrogenesis (Fig.

In other words, the dendritic, spike-dependent manner in which tuft input changes the input-output relationship between basal input and frequency output is explicitly accounted for in the form of the composite equation.

We create three abstract models to describe the input-output relationship from tuft and basal excitatory input to firing rate output.

The Phase Response Curve (PRC), a simple input-output characterization of single cells, can provide insights into individual and collective properties of neurons and networks, by quantifying the impact of an infinitesimal depolarizing current pulse on the time of occurrence of subsequent action potentials, while a neuron is firing tonically.

Recently, key results from the mathematical theory of coupled oscillators sparked a lot of interest: a simple input-output characterization of the units composing a network, known as their phase response (or phase resetting) carve (PRC), is sufficient to classify and predict individual and collective properties.

We employed the current-clamp configuration and focused on the input-output relationship between the phase shift Ag), induced in the cell firing cycle by an external current pulse, and the phase (p at which the pulse was repeatedly delivered (Fig.