Neutral Theory and Scale-Free Neural Dynamics

Neural tissues have been consistently observed to be spontaneously active and to generate highly variable (scale-free distributed) outbursts of activity in vivo and in vitro. Understanding whether these heterogeneous patterns of activity stem from the underlying neural dynamics operating at the edge of a phase transition is a fascinating possibility, as criticality has been argued to entail many possible important functional advantages in biological computing systems. Here, we employ a well-accepted model for neural dynamics to elucidate an alternative scenario in which diverse neuronal avalanches, obeying scaling, can coexist simultaneously, even if the network operates in a regime far from the edge of any phase transition. We show that perturbations to the system state unfold dynamically according to a “neutral drift” (i.e., guided only by stochasticity) with respect to the background of endogenous spontaneous activity, and that such a neutral dynamics—akin to neutral theories of population genetics and of biogeography—implies marginal propagation of perturbations and scale-free distributed causal avalanches. We argue that causal information, not easily accessible to experiments, is essential to elucidate the nature and statistics of neural avalanches, and that neutral dynamics is likely to play an important role in the cortex functioning. We discuss the implications of these findings to design new empirical approaches to shed further light on how the brain processes and stores information.

Popular Summary

The mammalian brain is perpetually in a state of electrochemical activity. Cascades of this activity pervade the neural tissue on multiple time scales, especially in the absence of any specific task or stimulus. Evidence suggests that this activity is not random but structured, and it may be critical to brain functions such as optimal information transfer, information storage, and sensitivity to stimuli. Researchers have conjectured that this activity occurs because the cortex operates close to the critical point of a phase transition. Here, we identify an alternative scenario to understand scale-free cortical avalanches in theoretical models of the cortex within the optics of neutral theories.

Neutral theories state that most biological evolution is caused by random drifts in the relative frequencies of neutral gene variations (ones that do not affect an organism’s ability to survive) in a population. Similar neutral approaches have proven illuminating to rationalize the proliferation of stem cells into heterogeneous populations, the emergence of diverse epidemic outbursts, and even the fate of memes in social networks.

Using well-accepted computational theoretical models of neural dynamics, we show that input signals unfold causally into the background of endogenous activity following a neutral drift. Consequently, scale-free causal avalanches are generated even if the system is not critical, thus providing a much more flexible and robust scenario for scale-free dynamics.

We elucidate the origin, relevance, and consequences of such neutral neural dynamics and discuss their far-reaching implications for shedding light on how the cortex processes and stores information.

Figure 1

Numerical integration of the model of Millman et al. [44] with $N=300$ neurons. (a) Phase diagram of the mean firing rate as a function of the synaptic strength parameter $w_{\mathrm{in}}$. For low values of $w_{\mathrm{in}}$, the stable state is a quiescent state with very low levels of activity (down state), whereas for large values of $w_{\mathrm{in}}$, the system exhibits high levels of activity (up state). Both states coexist for intermediate strength values (shaded region), allowing for up-and-down transitions. Importantly, the transition is discontinuous. (b) Time series of the network firing rate for $w_{\mathrm{in}}=50\mathrm{pA}$ illustrate the system’s bistability, with eventual (stochastic) jumps between up and down states. (c) Raster plot (for the same times as above) in which distinct colors are used for different causal avalanches. (d) Raster plot zoom (broken lines) demonstrating the intermingled and temporally overlapping organization of different causal avalanches. Model parameters have been set as in Ref. [44] (see Appendix pp2).

Figure 2

Probability distribution functions (PDF) for the avalanche size and durations within the up-state phase in the model of Millman et al. [44] using two different methods (double logarithmic plot). (a) Causal avalanches were defined using the same criterion as in Ref. [44], for several values of the external input $f_e$, confirming the observation that sizes and durations are power-law distributed with the same exponents of an unbiased branching process, i.e., $\tau =3/2$ and $\alpha =2$, respectively [9, 11]. (b) “Time-correlated” avalanches, defined with the standard temporal binning method [8] (which estimates causality by temporal proximity), using five different time intervals $\mathrm{\Delta }t$ to bin the data, including one coinciding with the average interevent interval (IEI) as usually done in the analyses of empirical data [8], for $f_e=5\mathrm{Hz}$; in this case, distributions do not obey a power-law distribution but have a characteristic scale. In all cases, simulations were performed in a network of $N=3000$ neurons (model parameters as in Ref. [44]; see Appendix pp2).

Figure 3

Causal avalanches in a minimal model for propagation activity, defined as cascades of events initiated from the spontaneous activation of one unit, without overlap between avalanches (i.e., a given node cannot be simultaneously part of more than one avalanche). (a) The activity of each avalanche is defined as the density of active elements in the system belonging to that avalanche, identified with different colors in the plot. The global activity density is represented with the gray colored line. Parameters of the model are taken deep inside the active phase, $\lambda =2$, $\mu =1$, for a system size $N={10}^4$ and small spontaneous activation rate $\epsilon ={10}^{-3}$. Whereas the global activity exhibits slight fluctuations around its steady-state value ${\rho }^{*}\simeq 1-\mu /\lambda$ (represented by the dashed line), individual avalanches can exhibit wild variability. (b) Probability distribution functions (PDF) for avalanche sizes and durations for different values of $\epsilon$ [other parameters as in (a), i.e., deep inside the active phase]. Avalanche statistics exhibit robust power-law scaling—limited by system size—with the same exponents of the neutral theory for avalanche propagation (marked with dashed lines for comparison).