Self-organized criticality for dendritic readiness potential

Self-organized criticality is a principle explaining avalanche-like phenomena obeying power-laws in integrate-and-fire type dynamical systems. Here, we demonstrate that the behaviorally relevant brain neurons, mediating voluntary and reflexive behaviors in crayfish, show signatures of self-organized criticality. The dendritic activities are into the critical states following power-laws; characterized by scaling functions within neurons; but mapped to the common scaling relation across individuals. The relation is in line with the extracellular neuronal avalanches in vertebrate species which have provided evidence of the critical brain hypothesis. Our intracellular data extend the universality of the hypothesis. In particular, it indicates that the pre-movement buildup activity before the voluntary behavioral onset"from crayfish to human", readiness potential, is shaped by the avalanches. The nervous systems can exploit the universal dynamics for volition across the phylogenetic tree.


Introduction
Voluntary initiation of behavior is widespread in the animal kingdom. In neurobiology, the readiness potential (RP) that precedes the initiation of a voluntary behavior can be observed in humans [18][19][20], rats [21], birds [22], fish [23], and crayfish [10,24]. Despite the diversity and various recording methods, their temporal profiles are quite similar in those species [18]. This suggests a general, scale-invariant physical mechanism underlying the RP. Here, we argue that self-organized criticality (SOC) [1,2] is the mechanism. SOC is a combined notion of criticality and self-organization. A critical state is far from equilibrium, like a steep slope of a sandpile proposed by Bak and colleagues [1,2]. Once the sandpile grows into a critical state, an addition of a single grain to any site can generate avalanches of all sizes obeying power-laws. No characteristic scales exist for the avalanches. Still controversial, they substantiate this idea as a cellular automaton model, termed sandpile (cf. our simulation, Fig. S1, Movie. 1, 2). It has provided a representative, fundamental conceptual framework to explain avalanche-like phenomena in physics, biology, and many other fields [3].
Neuronal avalanche, one example of the SOC behavior, has been found in several vertebrate nervous tissues using extracellular recordings [11,[14][15][16][17]. However, no work has demonstrated that the neuronal avalanche occurs in a single neuron on the dendritic processes, and no association of the neuronal avalanche with the readiness potential exists. In invertebrate species, a single neuron is large relative to the whole brain size, and can receive thousands of presynaptic, collective network activity as synaptic bombardments.
To test whether a real physical system shows the SOC behavior, there are four signatures required to be examined [13,25]; (1) multiple power-laws in duration, size, and spectrum of the activity; (2) exponents of the duration and size of avalanches satisfying the crackling noise relation [4,26]; (3) universal scaling function characterizing the scale-invariant shape of avalanches [27]; (4) self-organization demonstrated by the robustness against some initial settings of the system [13,28,29].
The crackling noise relation [4,26] is satisfied when the two exponents of size and duration of avalanches, α and τ , satisfy (τ − 1)/(α − 1) = γ, where γ is the slope of the log-log plot of the size and duration of the synaptic activity, and the exponent of the spectrum density S(f ) [30]. In our system, the crayfish brain, self-organization can be examined by robust achievements of the critical states against individual differences from identifiable neurons with natural variations of dendritic branching structures.
We show below that the brain neurons of crayfish act in critical states, generating neuronal avalanches on their dendritic processes and are self-organized. We compare the scaling relation with the BTW sandpile [1,2], the stochastic random walk model proposed for RP [18], and vertebrate species [12,13]. Dendritic neuronal avalanche in a readiness potential neuron. a, Intracellular recording from a resting crayfish. The treadmill has been described previously [24]. b, Definition of dendritic avalanche on the readiness potential neuron [10]. We defined an avalanche in terms of the size and duration of the synaptic summation. c, Upper trace: continuous dendritic neuronal avalanches of the readiness potential neuron. Spike activity is suppressed by current injection (-1 nA). The trace has been detrended, and the avalanches are detected over the broken line. Lower traces: 25 examples are indicated. d-f, Power-law fits to the sizes d and durations e of the avalanches. f, Comparison of data with the prediction on the crackling noise relation [4,26].

Results
The crayfish were placed on the spherical treadmill system [24] and voluntarily initiated, continued, and terminated walking (Fig. 1a). Intracellular recording from a RP neuron was obtained (Fig. 1bc, Fig. S1). We analyzed the activity when the animals were silent in a dark room; thus, we did not give any specific external stimuli. The synaptic activity from an RP neuron under minus 1 nA current injection, was detrended by linear subtraction (Fig. 1bc). The avalanches (N = 4427) were defined by summation of the synaptic activity over the broken line (Fig. 1c). The duration and size of each avalanche was defined from the transient activity (Fig. 1b). Power-law fitting of the size and duration for the histogram was performed (Fig. 1de), and the predictions of the scaling theory [4,26,30] were examined (Fig. 1f). The log-log plot of the duration and  [10]. c, Power spectrum of the neuron active when the crayfish stops walking [10]. d, Averaged spectrum of 112 neurons from different individuals and that of leg electromyography. Valley at 50 Hz is an artifact caused by the noise cut filter. Fifty and 90 percent of data are represented by the ribbon. e, Distribution of the slopes of the power-law fits.
size of the synaptic avalanches was in line with the prediction. Thus, the strict criterion for the criticality was satisfied. We compared the activity of RP and sandpile (Fig. S1), and confirmed that the sandpile model could shape RP like, buildup activity ( Fig. S2) similar to the readiness potential of this neuron [10].
To examine the power spectral density S(f ), we analyzed the membrane potential activity of 112 brain neurons sometimes with intense spike activity ( Fig. 2a-d). The power spectra of the three descending brain neurons involved in voluntary initiation, continuation, termination of walking [10] and other brain neurons were analyzed. The membrane potential before the behavioral onset was analyzed (Fig. 2a). The power spectrum exhibited almost 1/f fluctuation. The excitatory synaptic summations were observed during the 20 sec before the onset (Fig. S3ab). About one second before the onset, an increase of synaptic fluctuation was observed, shaping the RP (Fig. S3b). We here newly observed that the ongoing spontaneous synaptic activity, including RP, showed one signature of SOC: 1/f fluctuation. The relation S(f )∼f −γ [30] was closely satisfied (Fig. 1f, Fig. 2a). The continuation neuron (C), which was highly active during walking [10], showed a straight line in the spectrum (Fig. 2b) of the activity at resting (Fig. S3cd). The termination neuron (T), showed near 1/f fluctuation, but there was no specific activation before the behavioral onset (Fig. S3ef). In either neuron involved in initiation, continuation and termination of voluntary walking, the relations were reasonably satisfied. Overall, we performed the power-law fits to 112 brain neurons from different individuals. While the leg electromyography did not show the 1/f fluctuation within the frequency range (Fig. 2d), the slopes of the neurons were distributed around -1 (Fig. 2e); thus, the brain as a whole appears in the state S(f )∼1/f . To further characterize the temporal profile of the dendritic avalanches, we attempted to extract the universal scaling function of each neuron [26] (Fig. 3). We scaled the activity along the time and size axes (Fig. 3ab). Dendritic neuronal avalanches were obtained from the filtered traces greater than 1 Hz to remove the slow drift. If the shape of the avalanches would be self-similar, we would be able to extract the universal scaling function in the sense that the shape is transformation invariant (Fig. 3a). The functions of the three functional neurons showed different shapes, but were consistent within each neuron for four different classes of the duration of avalanches (Fig. 3c). Therefore, neurons exhibited diversity of the dynamics in terms of the avalanche between neurons, but each neuron showed consistent scaling curves within itself. However, the mechanism that determines the shape of the function is unknown. The dendritic structure may play an important role for the self-similar dynamics.
The universality of the scaling was examined by comparing the scaling relation of crayfish with that of other vertebrates and models [13]: sandpile model [1,31]; random walk model for human RP [18]. We compared the relation within and among individual neurons including identified neurons in crayfish (Fig. 4). Forty nine out of 76 neurons satisfied the crackling noise relation from different individuals including one non-spiking giant interneuron (NGI) [8], five continuation types (C) [10], three medial giant neurons (MG) [5] all of which can be simply described as the single line estimated by the linear regression, indicating the shared critical dynamics (Fig. 4a). The slope of the line (1.35), was found to be reasonably close to that of rat and other species (1.28) [13]. The models except for the random walk model were critical.
The remaining 27 neurons including two C neurons were not critical. This study had three limitations that must be considered: the recording capability of synaptic activity might have been limited to local regions; the neurons might have been locally sub-critical; the criticality might have had individual differences. The RP neuron is in green. The relation changes position spontaneously and upon current injection, but constrained to the linear relation. The exponents of models are, sand : our estimates of the sandpile; sand2 : estimation by [31]; random walk : stochastic leaky integrator model for the human RP [18]. MG: medial giant neuron [6]. NGI: non-spiking giant neuron [8]. C: continuation type neuron [10]. b, Stimulus-response type, feed-forward network view, such as the network of the escape behavior of crayfish [6]. c, Self-organized type, recurrent network view of this study. The feed-forward network is embedded within it.

Discussion
The hierarchical feed forward network view (Fig. 4b) is often taken in behavioral neurobiology. For example, the reflexive circuits for escape behaviors centered by the command neurons (Fig. 4b) have been well studied in crayfish [5,9]. The circuit is often described as feed-forward. However, the circuit has the recurrent pathway to suppress the reafference by the consequence of self-movement [7]. In addition, we have shown previously that the circuit for voluntary walking is organized recurrently [10]. Thus, the basic architecture of the circuit can be considered as a recurrent type (Fig. 4c). We have shown that the scaling behavior of the spontaneously generated avalanches is shared not only with the neurons involved in the voluntary behavior, but also reflexive interneurons: NGI for equilibrium response [8]; MG for escape response [6]. Therefore, the reflexive circuits appear to be not parallel to voluntary ones, but embedded within the entire recurrently connected network (Fig. 4c).
Our results demonstrate that the crayfish nervous system acts in a critical state. In general, a self-organized critical system has susceptibility and scale-invariance in the avalanches [3]. The properties can explain the readiness potential. Consider the BTW sandpile model (Fig. S2b) [1][2][3][4] instead of a neuron. If our measurement capability is limited, we cannot recognize when and where a tiny single grain is added; we can not identify the cause of an avalanche. In that sense, susceptibility can be regarded as spontaneity. Also, we cannot capture the whole states of the sandpile so that we can not predict the size and duration of an avalanche. Even if we can track back a single history of a large avalanche, no measures can be made for the averaged avalanche because of the lack of scales. Indeed, we can construct a readiness potential from the sandpile avalanches simply using the thresholding mechanism (Fig. S2b) which is the basic mechanism of SOC [4]. However, there is no specific time point when the readiness potential starts to build up if the potential is shaped by the scale-invariant avalanches. We consider that spontaneity of the readiness potential as susceptibility and scale-invariance in the avalanches through the way of understanding of SOC.
Altogether, such properties in SOC without fine parameter tuning can be considered as a default setting of the nervous systems for both reflexive and voluntary initiations of behaviors. For the escape response, the embedded circuit (Fig. 4b) should have a higher threshold to generate action potential as a command through the specific sensory pathway, but the sub-threshold critical dynamics is likely to affect the response variability. Thus, only presynaptic activity is critical, but output spike is extremely rare. Only strong stimuli to the head or tail allow the command neurons to generate an action potential. These hard-wired pathways have been shaped by evolutionary processes [9]. For the voluntary initiation of behavior, the circuit can exploit the default, avalanche dynamics without any specific sensory nor any single trigger neuron. The presynaptic activity is critical and the output spike activity is functional in the readiness discharge neuron [10,24]. The nervous system is full of noise and fluctuation. The system is susceptible to minor fluctuation, such as thermal noise within the nervous system [32] that can emit a readiness potential signaling an onset of voluntary behavior [10,24].
We conclude in broader contexts. In neurobiology, spontaneous behavior at the individual level has been described as power-law distributions [33][34][35]. For example, the power-law behaviors including the Lévy walks have been well known in Drosophila melanogaster [35] and they have been discussed in the context of behavioral ecology [35] and volition [34]. It would be fruitful to compare our results with unidentified neural implementations for the voluntary behavior in the fly. The observation of the neuronal avalanche is rare in the invertebrate system except for the leech [36]. Furthermore, designing a volition is another path. In material science, neuromorphic devices have been attempted to mimic the neuronal avalanche [37,38]. Because of the universality of the mechanism of SOC, evolution would have favored the natural dynamics for shaping the readiness potential across the evolutionary tree [18]. This finding can give an implication of making an artificial spontaneity for an autonomous agent. postsynaptic to visual interneurons. Journal of Comparative Physiology A 162 (6), 705-714 (1988) .
showing only total area of each avalanche caused by single grains added on a cell randomly chosen, starts from subcritical state and grows into critical states where system-wide large avalanches occur in the later part "spontaneously" (in the sense that an observer can not see a tiny grain) and "unpredictably" (in the sense that an observer can not capture all of the states and do not know the seed and algorithm of the pseudo random number generator). Fig. S1. Comparing readiness potential and sandpile dynamics, scaling function, crackling noise relationship. a, b, d, These are the same with the main figure (Fig. 1c-f), but shown for the comparison. a, Readiness potential neuron with the glass microelectrode. b, Membrane potential activity with the current injection (-1 nA) to suppress the action potentials. c, The synaptic avalanches were divided into 4 classes using 35, 50, and 80 percentiles of duration. The classes are into four columns, and the scaling process is visualized in three rows (top: original avalanches, middle: scaled by duration, bottom: scaled by size). The density of the overdrawn avalanche trajectories is visualized by the color scale. The high-density green part represents the averaged avalanche shape. The averaged trajectories are similar, thus we successfully extracted the scaling function. d, Power-law fittings for size (left), duration (middle). The exponents satisfied the crackling noise relationship (right). e, One snapshot of the toppling sandpile. f, The avalanche dynamics of the sandpile. Input sand grains were added when each avalanche terminated, to achieve the complete time-scale-separation. The y axis represents the number of over critical cells. g, The scaling process is the same as c. h, The crackling noise relation of the sandpile.  Fig. S1f. b, Readiness-potential-like ramping activity can be shaped using the avalanches. The triggered pointspresumed behavioral onsets -were set at the 99 percentile of the largest activation of the avalanches. c, Spectral analysis of the traces, confirming the exponent is almost 1.58 [1], and 1.57 in Fig. S1h.  Fig. S3. Morphology and membrane potential dynamics of brain neurons involved in voluntary initiation, continuation and termination. a,c,e, The morphological characteristics of the neurons reported previously [10]. b,d,f, The intracellular membrane potential dynamics of the neurons reported previously [10], we are focused on the dynamics particularly 20 sec prior to the voluntary initiation of walking. The traces are from each individual and aligned with the time of the initiation. b, The readiness potential can be observed in the four of five traces. These five traces were selected from 18 ones [10]. d, This continuation type of neurons was recorded from seven individuals (denoted as C in Fig 4a). This activity was so intensive that many action potentials were observed. Although they were named as continuation type, this activity showed readiness increase prior to the voluntary onset. f, This termination type of neurons was active when the animal stopped walking. The dynamics does not change prior to the voluntary onset, but active subthreshold dynamics can be observed.