Local homeostatic regulation of the spectral radius of echo-state networks

Abstract

Recurrent cortical network dynamics plays a crucial role for sequential information processing in the brain. While the theoretical framework of reservoir computing provides a conceptual basis for the understanding of recurrent neural computation, it often requires manual adjustments of global network parameters, in particular of the spectral radius of the recurrent synaptic weight matrix. Being a mathematical and relatively complex quantity, the spectral radius is not readily accessible to biological neural networks, which generally adhere to the principle that information about the network state should either be encoded in local intrinsic dynamical quantities (e.g. membrane potentials), or transmitted via synaptic connectivity. We present two synaptic scaling rules for echo state networks that solely rely on locally accessible variables. Both rules work online, in the presence of a continuous stream of input signals. The first rule, termed flow control, is based on a local comparison between the mean squared recurrent membrane potential and the mean squared activity of the neuron itself. It is derived from a global scaling condition on the dynamic flow of neural activities and requires the separability of external and recurrent input currents. We gained further insight into the adaptation dynamics of flow control by using a mean field approximation on the variances of neural activities that allowed us to describe the interplay between network activity and adaptation as a two-dimensional dynamical system. The second rule that we considered, variance control, directly regulates the variance of neural activities by locally scaling the recurrent synaptic weights. The target set point of this homeostatic mechanism is dynamically determined as a function of the variance of the locally measured external input. This functional relation was derived from the same mean-field approach that was used to describe the approximate dynamics of flow control.The effectiveness of the presented mechanisms was tested numerically using different external input protocols. The network performance after adaptation was evaluated by training the network to perform a time delayed XOR operation on binary sequences. As our main result, we found that flow control can reliably regulate the spectral radius under different input statistics, but precise tuning is negatively affected by interneural correlations. Furthermore, flow control showed a consistent task performance over a wide range of input strengths/variances. Variance control, on the other side, did not yield the desired spectral radii with the same precision. Moreover, task performance was less consistent across different input strengths.Given the better performance and simpler mathematical form of flow control, we concluded that a local control of the spectral radius via an implicit adaptation scheme is a realistic alternative to approaches using classical “set point” homeostatic feedback controls of neural firing.Author summaryHow can a neural network control its recurrent synaptic strengths such that network dynamics are optimal for sequential information processing? An important quantity in this respect, the spectral radius of the recurrent synaptic weight matrix, is a non-local quantity. Therefore, a direct calculation of the spectral radius is not feasible for biological networks. However, we show that there exist a local and biologically plausible adaptation mechanism, flow control, which allows to control the recurrent weight spectral radius while the network is operating under the influence of external inputs. Flow control is based on a theorem of random matrix theory, which is applicable if inter-synaptic correlations are weak. We apply the new adaption rule to echo-state networks having the task to perform a time-delayed XOR operation on random binary input sequences. We find that flow-controlled networks can adapt to a wide range of input strengths while retaining essentially constant task performance.

A devil's advocate view on 'self-organized' brain criticality

“…Critical, or near-critical brain dynamics is in this perspective nothing more than a particular aspect of the demand to retain stationary functionalities, here the resting state activity. This does not rule out secondary advantages of operating close to second-order phase transition, like improved information processing [2,3].…”

“…holds as a suitable time averages. Given a target spectral radius R w , neurons can satisfy the flow condition (7) by regulating their gain [3], which corresponds to a rescaling of the afferent synaptic weights. This is quite remarkable, as it implies that a global quantity, the spectral radius, can be regulated by relying solely on local information.…”

“…This is quite remarkable, as it implies that a global quantity, the spectral radius, can be regulated by relying solely on local information. The adaption rule resulting from (7), which has been termed 'flow control' [3], is effectively based on the circular law of random matrix theory, which states that the spectral radius of a matrix with uncorrelated entries is given by the variance of its elements [27]. The circular law states also that the eigenvalues of a real but nonsymmetric matrix Ŵ are uniformly distributed in the complex plain on a disk with radius R w .…”

“…This implies that most eigenvalues are smaller in magnitude. Note that flow control is an online rule, functioning while the system is operative, viz while processing a continuous stream of external inputs [3].…”

Local Homeostatic Regulation of the Spectral Radius of Echo-State Networks

“…The authors acknowledge the financial support of the German research foundation (DFG) and discussions with R. Echeveste. This manuscript was published as a pre-print on biorxiv (Schubert and Gros, 2020 ).…”