Firing rate predictions in optimal balanced networks
How are firing rates in a spiking network related to neural input, connectivity and network function? This is an important problem because firing rates are a key measure of network activity, in both the study of neural computation and neural network dynamics. However, it is a difficult problem, because the spiking mechanism of individual neurons is highly non-linear, and these individual neurons interact strongly through connectivity. We develop a new technique for calculating firing rates in optimal balanced networks. These are particularly interesting networks because they provide an optimal spike-based signal representation while producing cortex-like spiking activity through a dynamic balance of excitation and inhibition. We can calculate firing rates by treating balanced network dynamics as an algorithm for optimising signal representation. We identify this algorithm and then calculate firing rates by finding the solution to the algorithm. Our firing rate calculation relates network firing rates directly to network input, connectivity and function. This allows us to explain the function and underlying mechanism of tuning curves in a variety of systems.
A basic question in neuroscience concerns the relation 
      between network structure and function. A major difficulty in relating the 
      two is multi-leveled complexity of neural systems in terms on 
      nonlinearity, variability and stochasticity. The present is concerned with 
      networks of spiking neurons which attempt to optimally represent 
      time-varying input signals. The performance cost function comprises a 
      quadratic error function and a penalty for high activity. The authors show 
      how the natural network dynamics of a network of spiking neurons can be 
      described as minimizing this cost function, leading to a so-called 
      balanced state, which is, arguably, widely regarded as a natural state for 
      biological neuron dynamics. An important attribute of the present 
      formulation is that it does not rely on linear approximations, as has 
      often been used in the past, leading to a more faithful representation of 
      the true dynamics. Within this framework the authors compute the firing 
      rates as the solution of a quadratic problem subject to non-negativity 
      constraints on the rates, allowing them to express the solution in closed 
      form as piece-wise linear function. They also relate the types of tuning 
      curves obtained (monotonic or bell-shaped) to the nature of the input, and 
      show that optimal performance can be achieved using heterogeneous tuning 
      curves, as has been, indeed, observed in biological systems. Overall I 
      found this paper interesting and insightful, and, in combination with ref. 
      [13] that deals with synaptic plasticity, constitute an original 
      contribution to the structure-function relationship. Specific 
      comments: 1) The form of E in (6) suggests that W must be symmetric 
      (since the anti-symmetric component of W does not contribute to the 
      quadratic term). However, eq. (7) itself does not seem to require 
      symmetry. Please relate to this issue. 2) Equation (6) suggests that 
      it must have a minimum at which dE/d\nu=0 so that \dot V = 0. However, why 
      does this entail V_i=0 (balanced state) rather than V_i=constant. 3) 
      Typo: Line 330, where $\beta I$ - something is missing.