by Silvia Scarpetta
Dipartimento di Fisica ‘E.R.Caianiello’, Università degli studi di Salerno, Via Giovanni Paolo II, Fisciano (SA), Italy & INFN gruppo coll di Salerno, Unità di Napoli, Italy
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Many experimental results, both in-vivo and in-vitro, support the idea that the brain cortex operates near a critical point, and at the same time works as  associative memory, with  a reservoir of multiple precise spatio-temporal patterns. However the mechanisms at the basis of these observations are still not clear. We study a model of spiking neurons, with recurrent connections that result from learning a set of spatio-temporal periodic patterns with a spike-timing dependent plasticity rule and a global inhibition. We investigate the ability of the network to store and selectively replay multiple spatio-temporal patterns of spikes, with a combination of spatial population and phase-of-spike code. After the learning stage, we study the dynamics of the network induced by a brief cue stimulation,  and we evaluate the storage capacity [1]. 
In absence of cue stimulation,  we study the spontaneous dynamics induced by noise. Notably when heterogeneity in neurons parameter is included in the model,  noise focusing can induce noise-initiated short replay and up/down alternation. Changing excitability we observe a non-equilibrium discontinous phase transition, that, at high value of noise produces a regime with alternation of up and down states, depending on the lifetime of the metastable states [2]. In this regime, critical features and  scale-free neural avalanches are observed [2,3]. The presence of both a non-equilibrium first order transition and the critical precursor phenomena in our model are crucially related to the interplay between noise and a structured connectivity which promotes collectivity. Criticality emerges naturally near the edge of the instability, in an associative memory network, with many metastable dynamical states. Notably the scaling relation between critical exponents of avalanches’ sizes and durations is satisfied.
Interestingly, there are some analogies with the discontinuous phase transition models that have been proposed for flocks to explain the scale-free critical features observed in flocks.

[1]  Information capacity of a network of spiking neurons,  S.Scarpetta Candia Physica A 2020,

[2] Hysteresis, neural avalanches and critical behaviour near a first-order transition of a spiking neural network, S.Scarpetta I. Apicella Candia,Phys. Rev. E 2018

[3] Critical Behavior and Memory Function in a Model of Spiking Neurons with a Reservoir of Spatio-Temporal Patterns. S.Scarpetta 2020 in “The Functional Role of Critical Dynamics in Neural Systems” Springer Series on Bio- and Neuro-systems, vol 11. Springer,

Many experimental results, both in-vivo and in-vitro, support the idea that the brain cortex operates near a critical point, and, at the same time, works as a reservoir of precise oscillatory spatio-temporal patterns, with both cue-induced and spontaneous reactivation of precise dynamical spatio-temporal patterns of spikes.
In this work we review recent results which link together memory functions, collective oscillations and critical behavior, in a model with leaky neurons (whose structured recurrent connectivity comes from learning multiple spatio-temporal phase-coded patterns using a rule based on Spike Timing-Dependent Plasticity) in presence of a Poissonian noise distribution (modelling spontaneous neurotransmitter release at individual synapses, as well as other sources of inhomogeneity and randomness that determine an irregular background synaptic noise).
We find that, depending on the excitability of the network, different working regimes are possible. For high excitability, collective oscillatory activity that replays one of the stored patterns emerges spontaneously, while for low excitability, there’s uncorrelated poissonian low activity. Between these two regimes, there is a critical region with bimodal rate distribution and UP/DOWN alternation. Notably, in this critical region, the avalanche size and duration distributions follow power laws. Critical exponents are consistent with a scaling relationship observed recently in neural avalanches measurements (W.L. Shew, W.P. Clawson, J. Pobst, Y. Karimipanah, N.C. Wright, and R. Wessel, “Adaptation to sensory input tunes visual cortex to criticality,” Nat. Phys. 11, 659 (2015))

Looking at firing rate and normalized variance during spontaneous dynamics while sweeping excitability H0 at fixed α, one observes a peak of normalized variance near the transition between low (uncorrelated) activity and high (collective) activity state. However very different behaviours are observed depending on the value of the noise.
Depending on the noise, and size of the system, one observes hysteresis (low noise or large size) or up/down alternation (high noise or small size). While at low noise (alpha < alpha_c) the transition in H is abrupt and hysteretic, as one approachs the critical noise alpha_c at critical excitability H0 the transition seems to be continuous and with avalanches of all sizes.
This remembers the Random Field Ising Model of Sethna et al PRL 1993 where, sweeping the external field through zero, one sees that approaching the critical value of randomness Rc, the first order transition becomes continuous, with scale-free avalanches.

Finally we study the model without noise (alpha=0) as a model of associative memory of collective oscillatory patterns, that code information through both WHO fires and WHEN fires. We see that storage capacity (max number of patterns successfully selectively retrievable) of such network is maximal at some values of Excitation Inhibition values, while critical fading of activity is seen at critical values of E and I. We compute maximal storage capacity and we see that a mixed code where not all units are active in the oscillatory pattern (some are oscillatory and the others are silents) is more efficient then a code where all units partecipate firing in the collective oscillation.

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