  Recurrent cortical networks provide reservoirs of states that are thought to
play a crucial role for sequential information processing in the brain.
However, classical reservoir computing requires manual adjustments of global
network parameters, particularly of the spectral radius of the recurrent
synaptic weight matrix. It is hence not clear if the spectral radius is
accessible to biological neural networks.
  Using random matrix theory, we show that the spectral radius is related to
local properties of the neuronal dynamics whenever the overall dynamical state
is only weakly correlated. This result allows us to introduce two local
homeostatic synaptic scaling mechanisms, termed flow control and variance
control, that implicitly drive the spectral radius towards the desired value
under working conditions.
  We demonstrate the effectiveness of the two adaptation mechanisms under
different external input protocols and the network performance after adaptation
by training the network to perform a time-delayed XOR operation on binary
sequences. As our main result, we found that flow control reliably regulates
the spectral radius for different types of input statistics. Precise tuning is
however negatively affected when interneural correlations are substantial.
Furthermore, we found a consistent task performance over a wide range of input
strengths/variances. Variance control did however not yield the desired
spectral radii with the same precision, being less consistent across different
input strengths.
  Given the effectiveness and remarkably simple mathematical form of flow
control, we conclude that self-consistent local control of the spectral radius
via an implicit adaptation scheme is an interesting and biological plausible
alternative to conventional methods using setpoint homeostatic feedback
controls of neural firing.
