Biological Plausibility of FORCE training In "Supervised Learning in Spiking Neural Networks with FORCE Training" by Wilten Nicola and Claudia Clopath. The authors create a learning rule for learning non-linear dynamics from populations of spiking neurons. The learning rule only seems to depend on:


an error signal
the firing rates of each neurons
an approximate inverse correlation matrix calculated from the firing rates of each neuron


Is it possible to calculate the inverse correlation matrix of using another population of neurons or is there some biological mechanism that could be used to explain this? Additionally, is this calculation sensitive to delay? Often, forgetting to take delay into account is the undoing of many learning rules and cognitive architectures.
From the paper:


  Although FORCE trained networks have dynamics that are starting to
  resemble those of populations of neurons, at present all top-down
  procedures used to construct any functional spiking neural network
  need further work to become biologically plausible learning rules
  [Sussillo and Abbott, 2009, Boerlin et al., 2013, Eliasmith et al.,
  2012]. For example, FORCE trained networks require non-local
  information in the form of the correlation matrix $P(t)$. However, we
  should not dismiss the final weight matrices generated by these
  techniques as biologically implausible simply because the techniques
  are themselves biologically implausible. More work should be done in
  implementing either FORCE, NEF, or spike-based coding networks using a
  biologically plausible learning mechanism based on synaptic plasticity
  or homeostasis [Bi and Poo, 1998, Pfister and Gerstner, 2006, Clopath
  et al., 2010, Graupner and Brunel, 2012, Babadi and Abbott, 2016,
  Vogels et al., 2011]. This has been resolved for spike-based coding
  networks and linear dynamical systems for example [Bourdoukan and
  Deneve, 2015]


Basically, calculating the correlation matrix $P(t)$ requires every neuron to know what every other neuron is doing, so the FORCE algorithm isn't biologically plausible. I'm not sure what alternative there would be calculating the correlation matrix. Maybe there's some mathematical way to approximate it gradually over time?