## Abstract

The extreme complexity of the brain has attracted the attention of neuroscientists and other researchers for a long time. More recently, the neuromorphic hardware has matured to provide a new powerful tool to study neuronal dynamics. Here, we study neuronal dynamics using different settings on a neuromorphic chip built with flexible parameters of neuron models. Our unique setting in the network of leaky integrate-and-fire (LIF) neurons is to introduce a weak noise environment. We observed three different types of collective neuronal activities, or phases, separated by sharp boundaries, or phase transitions. From this, we construct a rudimentary phase diagram of neuronal dynamics and demonstrate that a noise-induced chaotic phase (N-phase), which is dominated by neuronal avalanche activity (intermittent aperiodic neuron firing), emerges in the presence of noise and its width grows with the noise intensity. The dynamics can be manipulated in this N-phase. Our results and comparison with clinical data is consistent with the literature and our previous work showing that healthy brain must reside in the N-phase. We argue that the brain phase diagram with further refinement may be used for the diagnosis and treatment of mental disease and also suggest that the dynamics may be manipulated to serve as a means of new information processing (e.g., for optimization). Neuromorphic chips, similar to the one we used but with a variety of neuron models, may be used to further enhance the understanding of human brain function and accelerate the development of neuroscience research.

## 1  Introduction

One of the most important hypotheses in neuronal dynamics is the criticality hypothesis (Hesse & Gross, 2014; Beggs, 2008; Kello et al., 2010; Beggs & Timme, 2012; Shew & Plenz, 2013; Marković & Gros, 2014). The brain, a complex system with a tremendous number of elements, brings immense attention to its collective neurodynamical behavior and related firing activities (Chialvo, 2010). The criticality hypothesis states that the brain is poised at the critical boundary between two different dynamic states. Neuronal states residing in the critical boundary exhibit a pattern of activity known as neuronal avalanche (Beggs & Plenz, 2003, 2004; Benayoun, Cowan, van Drongelen, & Wallace, 2010), an intermittent aperiodic firing behavior of groups of neurons in which event sizes show no characteristic scale and are described by power laws. Experimental results and theoretical models in support of this criticality have been found to be rather ubiquitous across multiple species, including rats (Gireesh & Plenz, 2008), adult cats (Hahn et al., 2010), and nonhuman primates (Petermann et al., 2009; Yu et al., 2011), and using different measurements, such as electroencephalography (EEG) (Meisel, Olbrich, Shriki, & Achermann, 2013), magnetoencephalography (MEG) (Poil, Hardstone, Mansvelder, & Linkenkaer-Hansen, 2012; Palva et al., 2013; Shriki et al., 2013), local field potentials (LFP) (Plenz & Thiagarajan, 2007), and functional magnetic resonance imaging (fMRI) (Tagliazucchi, Balenzuela, Fraiman, & Chialvo, 2012; Haimovici, Tagliazucchi, Balenzuela, & Chialvo, 2013). Furthermore, the different characteristics of neuronal dynamics behavior can be used as biomarkers of mental diseases. For instance, fMRI studies have been used to evaluate various pathological conditions such as Alzheimer's disease (He, Chen, & Evans, 2008), schizophrenia (Garrity et al., 2007), and epilepsy (Laufs et al., 2007; Osorio, Frei, Sornette, Milton, & Lai, 2010) showing a characteristic deviation from a power law distribution. EEG data of epileptic dynamics show a supercritical state behavior that deviates from the power law statistics of normal brain dynamics (Hobbs, Smith, & Beggs, 2010; Meisel, Storch, Hallmeyer-Elgner, Bullmore, & Gross, 2012).

Although such criticality in neurodynamical behavior has been inferred in many scenarios, the debate about this criticality hypothesis is ongoing. First, it is hard to conceive that our brain, which is resilient to many perturbations, can operate on only a critical boundary. In fact, in the brain, the self-organization never precisely reaches the critical state since it is subject to continuous external simulations (Bonachela, De Franciscis, Torres, & Mũnoz, 2010; Priesemann et al., 2014). Second, it is unclear how individual neurons or synapses know the entire neural network has reached criticality. In the neural system, collective mechanisms have to be evaluated from an internal perspective, which likely relies on the dynamics of single neurons or synapses and their interactions (Hesse & Gross, 2014).

Recently, several solutions have been proposed to resolve this controversy. One possible way is to construct a brain-phase diagram that characterizes the behavior of neuronal complex systems into different phases (Brunel, 2000; Moretti & Muñoz, 2013). A number of recent studies demonstrate neuronal avalanche using numerical analysis and computer simulation. By tuning different parameters in the network, such as network connectivity or refractory time, they are able to find the phase transition of collective neuronal dynamics behavior and create a phase diagram (Juanico & Monterola, 2007; Taylor, Hartley, Simon, Kiss, & Berthouze, 2013; Hartley, Taylor, Kiss, Farmer, & Berthouze, 2014; Lee, Lopes, Mendes, & Goltsev, 2014; Moosavi & Montakhab, 2014). With the aid of brain-phase diagrams, one can easily tell whether the normal brain states reside on a critical point or boundary or in a phase region. In addition, a phase diagram that has both network information and individual neuron properties may be studied and accessed on how local information of neurons affects the entire network's dynamic behavior. However, in vivo, carrying out experiments is limited since it is hard to access control parameters (the parameters that need to be tuned for the phase transition) in experimental settings. Therefore, many of the studies are limited to the construction of formal mathematical models and computer simulations, which is time-consuming and not easily scaled up. The recent advent of advanced neuromorphic chips has made it possible to investigate neuronal dynamics, at least within the approximate model of neurons and their interconnections, more easily. Our neuromorphic simulation reduces the need for solving complex differential equations. It thus provides a much higher degree of flexibility and reduces computational and time costs. Furthermore, this kind of neuromorphic chip can be easily scaled up, and allows us to conduct experiments with great flexibility and to accelerate our understanding of neuronal dynamics with higher degrees of freedom that could not have been done before (Pfeil et al., 2013).

Several researchers however, have noticed the importance of noise in dynamic systems (Lee et al., 2014; Moosavi & Montakhab, 2014; Jung, Cornell-Bell, Madden, & Moss, 1998). Williams-Garca, Moore, Beggs, and Ortiz (2014), established the nonequilibrium phase diagram of the cortical-branching model (CBM) and discussed how noise affects the phase diagram. Their work is quite similar to ours except that they do not realize that noise plays an essential role for the brain dynamics. Recently, our theoretical work (Ovchinnikov, 2016; Ovchinnikov et al., 2016) showed that in the presence of noise, the phase diagram of a stochastic dynamical system consists of three major phases: thermal equilibrium (T-phase), noise-induced chaotic (N-phase), and ordinary chaos (C-phase). In other words, the sharp critical boundary between the T- and C-phases in the deterministic case becomes a full-dimensional N-phase with the introduction of the noise. The presence of the N-phase is of paramount importance for understanding the cognitive functions of a healthy brain as well as its well-being (Hudson, Calderon, Pfaff, & Proekt, 2014). In order to study and characterize these phases further, we used the neuromorphic chip Spikey to emulate the different neuronal dynamic phases and compare them with clinical data from the brain. We demonstrated in this study that under different control parameters of neuronal firing thresholds and external noise levels, the behavior of neuronal dynamics can be indeed categorized into three phases. To validate our results, we compared our outcomes with clinical recordings from human brain slices. The fact that we observed similar characteristics proved that our emulations, within the approximation of the model of the neuron, reproduced those of the biological brain. With the flexibility of the artificial neuron of the neuromorphic chip, we showed that the early self-organized critical boundary became a full-dimensional phase as induced by noise in the neuronal dynamics. The phase diagram could be constructed based on power spectrum analysis of neuronal recordings. Finally, we demonstrated another method of constructing the phase diagram using collective neuronal firing activities in the neural network. We also compared our work with other experimental and simulation studies from the literature and validated our proposition.

## 2  Settings

For the emulation study, the Spikey chip used a leaky integrate-and-fire (LIF) neuron model with a large set of accessible parameters for each individual neuron and synapse. In this study, we used the firing threshold voltage ($Vth$) as the controllable variable to tune the system dynamics. It is worth noting that unlike doing biological experiments on the actual brain, it is easy to reconfigure the neuronal network structure (by changing the synapse connection and synapse weights) on the Spikey chip. This capability provides us flexibility for future studies.

We constructed a recurrent network of neurons with sparse and random connections on the Spikey chip.

As illustrated in Figure 1, a group of LIF neurons shown in the brain was used. Each neuron was configured to have sparse connections that were randomly drawn from all other hardware neurons. The network was stimulated by an external input modeled as a uniform-time-distributed signal. In order to have no mean effect on neurons firing, we chose to have half of the external stimuli to be excitatory (green circles) and half of them inhibitory (red circles). The resting membrane potential of the neurons was chosen to be $-$65 mV, which slightly deviates from the biological neuron resting membrane potential (usually between $-$75 mV and $-$70 mV in vitro). In our study, the small deviation of the absolute membrane potential should not matter because only the difference between the neuronal firing threshold and the resting membrane potential is important. Because of the limit of hardware parameter range, we set the resting membrane potential at $-$65 mV (see also section A.1 for details).

Figure 1:

Illustration of emulation settings on the Spikey chip. A recurrent neural network constructed with 192 neurons is shown in the “brain.” The network was stimulated by external input signals, half of them excitatory (green circles) and half of them inhibitory (red circles). Two types of recordings were extracted from the emulation. The membrane potential V$m$ of a randomly chosen neuron can be recorded during the emulation, as shown in the bottom left. The raster plot at the bottom right shows “spike train” data, which record the spike events for every neuron in the recurrent network.

Figure 1:

Illustration of emulation settings on the Spikey chip. A recurrent neural network constructed with 192 neurons is shown in the “brain.” The network was stimulated by external input signals, half of them excitatory (green circles) and half of them inhibitory (red circles). Two types of recordings were extracted from the emulation. The membrane potential V$m$ of a randomly chosen neuron can be recorded during the emulation, as shown in the bottom left. The raster plot at the bottom right shows “spike train” data, which record the spike events for every neuron in the recurrent network.

Two types of recordings were extracted from the emulation. The membrane potential V$m$ of a randomly chosen neuron can be recorded during the emulation, as shown at the bottom left of Figure 1, in which the raster plot shown at the bottom right of Figure 1 is the “spike train” data, which are indicated with dots if the neurons are active at that time and unmarked otherwise.

## 3  Results

### 3.1  Emulation Results from the Neuromorphic Chip

In this section, we present three types of neuronal dynamics. In our emulation setup, the average time interval between two consequent external stimuli was set at 25 ms. By setting the neuron firing threshold to $-$62 mV, $-$60 mV, and $-$57 mV, respectively, we observed three different neuronal dynamic behaviors, as illustrated in Figure 2. The top row of Figure 2 shows the emulated network activities given by raster plots, and the bottom row includes the membrane potential recordings of a randomly chosen neuron. As shown in Figure 2a, we observed a constant oscillatory firing behavior. There was an intermittent avalanche firing behavior from the membrane potential recordings in Figure 2b and nearly no firing activity in Figure 2c. Based on these observations, we refer to these three typical dynamic behaviors as the seizure-like firing activity, the normal firing activity, and the coma-like firing activity with their corresponding phase states as the C-phase (chaotic phase), the N-phase (noise-induced phase), and the T-phase (thermal equilibrium phase), respectively (Ovchinnikov et al., 2016; Ovchinnikov & Wang, 2017; Ovchinnikov, 2016).

Figure 2:

Three typical neuronal dynamic behaviors in the neural network. The raster plots of spike train data and membrane potential recordings are shown in the top and bottom rows, respectively. (a) Constant oscillating firing behavior presents in the seizure state/C-phase with a high activity correlation. (b) Normal state/N-phase behavior shows intermittent firing events with relatively low activity correlation. (c) None or only a few firing activities are present in the coma state/T-phase with extremely low activity correlation.

Figure 2:

Three typical neuronal dynamic behaviors in the neural network. The raster plots of spike train data and membrane potential recordings are shown in the top and bottom rows, respectively. (a) Constant oscillating firing behavior presents in the seizure state/C-phase with a high activity correlation. (b) Normal state/N-phase behavior shows intermittent firing events with relatively low activity correlation. (c) None or only a few firing activities are present in the coma state/T-phase with extremely low activity correlation.

Although it seems hard to quantitatively distinguish the differences among those three states from the raster plots at a glance, as we will explain later, a correlation parameter can be established and used to categorize each phase. At this point, we discuss the raster plots qualitatively. In the coma-like state shown in Figure 2c, the firing activity is scarce from the emulation, and this firing activity does not propagate (no neuronal avalanche occurs). This “quiet” behavior is a consequence of the big potential gap between the neuron firing threshold and the resting potential; thus, the input noise is not enough to overcome the neuron threshold barrier. In the seizure-like and normal states in Figures 2a and 2b, however, the firing activities can be observed more often. In Figure 2b, the neurons fire intermittently, which, when compared to Figure 2c, is clearly discernable from the raster plot, while in Figure 2a, almost every neuron fires continuously.

### 3.2  Data from Brain Slice Recordings

To further validate our findings, in this section, we compare our emulation data with recordings from real brain tissues. For the latter, neocortical sample sites from pediatric surgery cases were excised for in vitro electrophysiological evaluation based on abnormal neuroimaging and electrocorticography (ECoG) assessments. Through electrophysiological and pharmacological methods, the three states can be replicated: the coma-like state was observed at the resting membrane potential in the deafferented slice, the normal physiological state can be induced by threshold membrane depolarization, and the seizure activity was induced by blocking GABA$A$ and GABA$B$ receptors with bicuculline (10 $μ$M) and phaclofen (6 $μ$M), respectively. Note that the small differences between two measurement settings for the brain slice experiments and the Spikey itself caused discrepancies in the results.1 As listed in Table 1, these differences include measurement sampling rate, recorded signal range, and dominant frequency related to different biological time constants. The experiment on the brain slices adopts a standard current clamp measuring technique. The measurement sampling rates are 100 $μ$s and 200 $μ$s in the emulation and brain slice recordings, respectively, which results in the observable highest frequency as 5 kHz and 2.5 kHz based on the Nyquist/Shannon sampling theorem. The recorded signal range ($y$-axis) is the membrane potential range. The dominant frequency represents the largest frequency components when both cases reside in the C-phase. In order to eliminate the differences of the data from the emulation and experiment in Table 1 and Figure 3, we used the FWHM (full-width half-maximum) of the typical firing activity time interval as the unit time to normalize the data. The amplitude of all the data for both sets was rescaled to 0 $∼$ 1 (see details in section A.3).

Figure 3:

Comparison between human brain slice data and emulation data in the coma-like (top row), the normal (middle row), and the seizure-like (bottom row) states. In the coma-like state, we did not observe any firing activities at both membrane potential recordings (see top row in panel a). Their corresponding power spectrum shows a sharp drop at low frequencies and the typical normalized power spectrum in the frequency domain is below 10$-3$ in the frequency domain of measurement (see top row, panel b). In the normal state, we observe the intermittent firing activities in both systems (see the middle row, panel a). Their power spectrum shows a 1/f$α$ (a linear line on log-log plot) behavior (see the middle row in panel b). In the seizure-like state, we observe the oscillatory behavior from both membrane potential recordings (see the bottom row in panel a) and find that the power spectra are both 1/f$α$ superimposed by peaks (see the bottom row in panel b).

Figure 3:

Comparison between human brain slice data and emulation data in the coma-like (top row), the normal (middle row), and the seizure-like (bottom row) states. In the coma-like state, we did not observe any firing activities at both membrane potential recordings (see top row in panel a). Their corresponding power spectrum shows a sharp drop at low frequencies and the typical normalized power spectrum in the frequency domain is below 10$-3$ in the frequency domain of measurement (see top row, panel b). In the normal state, we observe the intermittent firing activities in both systems (see the middle row, panel a). Their power spectrum shows a 1/f$α$ (a linear line on log-log plot) behavior (see the middle row in panel b). In the seizure-like state, we observe the oscillatory behavior from both membrane potential recordings (see the bottom row in panel a) and find that the power spectra are both 1/f$α$ superimposed by peaks (see the bottom row in panel b).

Table 1:
Comparison of the Parameters in the Two Measurement Systems: Emulation on Spikey and Experiment on Human Brain Slices.
 Emulation on Spikey Experiment on Brain Slice Measurement method — Current clamp recording Measured subject 1 of 192 neurons in Spikey neural network 1 neuron in human brain slice Measurement sampling rate $∼$100 $μ$s 200 $μ$s Observable highest frequency $∼$5 kHz 2.5 kHz Recorded signal range $-$75 mV $∼$$-$60 mV $-$120 mV $∼$ 60 mV Dominant frequency 0.1 kHz 0.02 kHz
 Emulation on Spikey Experiment on Brain Slice Measurement method — Current clamp recording Measured subject 1 of 192 neurons in Spikey neural network 1 neuron in human brain slice Measurement sampling rate $∼$100 $μ$s 200 $μ$s Observable highest frequency $∼$5 kHz 2.5 kHz Recorded signal range $-$75 mV $∼$$-$60 mV $-$120 mV $∼$ 60 mV Dominant frequency 0.1 kHz 0.02 kHz

Notes: The experiment on human brain slices used whole-cell patch clamp recordings in the current clamp mode; the measurement sampling rates are 100 $μ$s and 200 $μ$s, respectively, which result in the observable highest frequency as 5 kHz and 2.5 kHz based on the Nyquist/Shannon sampling theorem. The recorded $y$-axis is the recorded membrane potential range for each measurement. The dominant frequencies represent the largest frequency components when both systems reside in the C-phase.

To further quantify the three states, we applied a power spectrum analysis for the normalized data from both emulation and the membrane potential recordings of the brain slices for the three phases. As expected, the power spectrum behaved quite differently in the frequency domain, which provided us with an indicator to analyze the dynamic system. Figure 3 shows the comparison of both clinical and emulation results. We again found similar traits in all three states. In the coma-like state (see the top row in Figure 3a), we did not observe any firing activity at membrane potential recordings in both clinical and emulation settings. Their corresponding power spectra showed a sharp drop at low frequencies, and the typical normalized power spectra (in the top row of Figure 3b) in the frequency domain are below 10$-3$. In the normal state (in the middle row of Figure 3a), we observed intermittent firing activities in both measurements. Their power spectra (in the middle row of Figure 3b) showed a 1/f$α$ behavior. In the seizure-like state (in the bottom row of Figure 3a), we observed an oscillatory behavior and found that the power spectra (the bottom row in Figure 3b) showed 1/f$α$ behavior superimposed by some peaks at low frequencies.

These results showed that the emulation provides a reasonable description of the activities in the real brain, at least within the approximation model we used. With the neuromorphic chip, we were able to further manipulate the dynamics in the N-phase by varying the parameters, indicating the N-phase has a width, related to the noise level as discussed below.

### 3.3  Constructing Brain Phase Diagram with Power Spectra

In this section, we demonstrate how the neuronal dynamics phase diagram is constructed using the Spikey emulation by controlling the firing threshold and the noise level. Owing to the high configurability of the chip, we can explore a relatively large controllable parameter space. Recently, our work on the approximation-free theory of neuronal dynamics (Ovchinnikov, 2016; Ovchinnikov et al., 2016) revealed that with noise, the early conceived self-organized criticality boundary becomes a general phase, and thus the phase diagram should consist of the three full-dimensional phases: the thermodynamic equilibrium phase (T-phase), the noise-induced phase (N-phase), and the ordinary chaos phase (C-phase) as described before (Ovchinnikov et al., 2016).

To validate this picture, we performed emulation studies on the Spikey chip. On the basis of the settings described in section 2, we also included the noise level as a controllable parameter. Specifically, the induced noise power intensity was controlled by the total stimuli number during the emulation period. The stimuli time of the noise follows a uniform distribution of a random number.

The resulting phase diagrams given on the plane of the firing threshold potential V$th$ and the noise intensity are illustrated in Figure 4a, and they were constructed through a power spectrum analysis of the membrane potential recordings, which were the same as the results in Figure 3. When no noise was present in the system, there were only two phases: the T-phase with no conspicuous dynamics and the C-phase featured by oscillatory chaotic activities. The zoomed Figure 4b reveals the sharp transition between these two phases: the C-phase featured by power spectra with equidistant peaks superimposed on the 1/f$α$ spectral density occurs at V$th$ = $-$60.6 mV, and the T-phase featured by no spiking activity at V$th$ = $-$60.5 mV. The transition was abrupt considering the fact that 0.1 mV is the smallest resolution for the Spikey chip; we could not get a sharper phase transition window. As the noise intensity increases, another distinct noise-induced phase (N-phase), as discussed earlier, emerges. The border of the N-phase was highly dependent on the noise intensity where a higher noise intensity results in a wider N-phase (in the sense of threshold range). From this simple emulation study, it is clear that the N-phase (which shows the intermittent avalanche firing activities and mimics a condition of wakefulness) should reside in a certain full-dimensional phase rather than at a critical boundary for a normal brain.

Figure 4:

Phase diagram of neuronal dynamics constructed from power spectrum analysis. (a) The phase diagram shows the dependence of noise versus threshold potential. In the deterministic limit (no noise present), the N-phase collapses onto a sharp boundary between the T-phase and C-phase (vertical dashed line at around $-$60.5 mV), as predicted by the previous theory. The noise-induced phase (N-phase) emerges when noise is present in the system. The noise is an essential parameter in this N-phase picture. The higher the noise intensity is, the wider the N-phase (in the sense of threshold range) becomes. The insets in each phase show the typical membrane potential behavior on different timescales. Zoomed panel b shows the sharp transition in the deterministic limit between the T-phase and the C-phase. A clear difference in both membrane potential recordings and their corresponding power spectra is seen, even though there is only a very small change from V$th$ = $-$60.6 mV to V$th$ = $-$60.5 mV in the case of when no noise is present.

Figure 4:

Phase diagram of neuronal dynamics constructed from power spectrum analysis. (a) The phase diagram shows the dependence of noise versus threshold potential. In the deterministic limit (no noise present), the N-phase collapses onto a sharp boundary between the T-phase and C-phase (vertical dashed line at around $-$60.5 mV), as predicted by the previous theory. The noise-induced phase (N-phase) emerges when noise is present in the system. The noise is an essential parameter in this N-phase picture. The higher the noise intensity is, the wider the N-phase (in the sense of threshold range) becomes. The insets in each phase show the typical membrane potential behavior on different timescales. Zoomed panel b shows the sharp transition in the deterministic limit between the T-phase and the C-phase. A clear difference in both membrane potential recordings and their corresponding power spectra is seen, even though there is only a very small change from V$th$ = $-$60.6 mV to V$th$ = $-$60.5 mV in the case of when no noise is present.

### 3.4  Constructing Brain Phase Diagrams with Mean Correlation

To further validate our results, in this section, we offer an alternative way to construct the phase diagram and make the comparison between the two versions of the phase diagram.

In order to take the entire neural network activity into account and give a fuller picture, we used an alternative way of analysis to reconstruct the brain phase diagram. Previous studies (Bornholdt & Rohlf, 2000; Bornholdt & Röhl, 2003; Meisel & Gross, 2009) have shown that it is advantageous to quantify the neuronal dynamics by defining a correlation parameter $〈Csync〉$ as the average over the correlations between pairs of neurons i, j, as
$Ci,j(τ)=1τ∫t0t0+τσi(t)σj(t)dt$
(3.1)
where $σi(t)$ is 1 if the neuron $i$ spiked at time $t$ or 0 otherwise. We can extract this parameter from the raster plots shown in the top row of Figure 2 (see the appendix for a detailed description of the method).

The 3D plot of the order parameter $〈Csync〉$ on the threshold-noise level plane is shown in Figure 5. For all noise levels, $〈Csync〉$ increases as the threshold decreases. At the threshold around $Vth$ = $-$60.5 mV, we see that $〈Csync〉$ becomes pronounced ($>10-3$)—a phase transition to the C-phase (red and pink). This is exactly the same results we observed for the phase transition based on the power spectrum analysis. When the noise intensity increased, a plateau, namely, the N-phase (green), shows up. The magnitude $〈Csync〉$ of the plateau in the middle at all noise levels was around $10-4$; the T-phase (in gray) without conspicuous firing activities usually has a correlation parameter below $∼10-6$, two orders of magnitude smaller.

Figure 5:

An alternative way to construct the phase diagram. Based on the correlation parameter $〈Csync〉$ (detailed in the text), a brain phase diagram was constructed similar to that in Figure 4. A 3D plot of $〈Csync〉$ in the threshold-noise-level plane is shown in this figure. For all noise levels, $〈Csync〉$ increases as the threshold decreases. At a threshold around V$th$ = $-$60.5 mV, we see that $〈Csync〉$ becomes pronounced ($>10-3$), that is, a phase transition to the C-phase, for all noise levels. When the noise intensity increases, we observed that a plateau, the N-phase, shows up. The magnitude of $〈Csync〉$ at the plateau for all noise levels is around $10-4$, which is depicted in green in the diagram, while the T-phase (in gray) without conspicuous firing activities usually has a correlation parameter below $10-6$, two orders of magnitude smaller.

Figure 5:

An alternative way to construct the phase diagram. Based on the correlation parameter $〈Csync〉$ (detailed in the text), a brain phase diagram was constructed similar to that in Figure 4. A 3D plot of $〈Csync〉$ in the threshold-noise-level plane is shown in this figure. For all noise levels, $〈Csync〉$ increases as the threshold decreases. At a threshold around V$th$ = $-$60.5 mV, we see that $〈Csync〉$ becomes pronounced ($>10-3$), that is, a phase transition to the C-phase, for all noise levels. When the noise intensity increases, we observed that a plateau, the N-phase, shows up. The magnitude of $〈Csync〉$ at the plateau for all noise levels is around $10-4$, which is depicted in green in the diagram, while the T-phase (in gray) without conspicuous firing activities usually has a correlation parameter below $10-6$, two orders of magnitude smaller.

Given the similarity of the phase diagrams derived from both the power spectrum method and the correlation parameter method, it is clear that the local neuron activity can reflect the dynamics of the global system. In other words, when the single neuron fires continuously, it yields a higher probability for the entire system to reside in the C-phase. Based on this knowledge, each neuron is affected by its network environment until the whole network settles down into a specific phase. To some extent, this fact can explain how the individual neuron or synapse and the local observation can be used to get information about some whole network properties, at least in part, and vice versa (Hesse & Gross, 2014).

## 4  Discussion

Clearly our current work is at very early stage. In our approach, the neuron model we used was quite simple. New neuron models discovered by neuroscientists at the cellular level may be incorporated into new neuromorphic chips to improve the emulation. It is noteworthy, that this work shows that it may be possible to emulate collective brain activity, which may deepen the understanding of brain function. This work also marks the beginning of a new era for brain simulation, inspired by the current rudimentary form of neuromorphic chips.

Clearly there are some substructures of the phase diagram for various brain conditions. Indeed, researchers using a pharmacological agent on the neuronal dynamics during a typical anesthesia cycle (Hudson et al., 2014) observed that the brain condition could be manipulated. As the concentration of the agent was gradually changed, the neuronal dynamics also gradually moved to consciousness (N-phase). The researchers also found that the recovery from a pharmacologically induced coma to consciousness travels through some discrete metastates, indicating that there may be additional subphases along with the brain dynamics path from the coma to its recovery. Another recent work supports phase transitions in brain activities (Torres & Marro, 2015). By means of computer simulation, they were able to show the different conditions or “phases” of the brain and transitions between them. They found that weak noise signals may serve to uncover the phase transitions during actual brain operation.

## 5  Conclusion

In this letter, we have studied neuronal dynamics of clinical recordings and emulations on a neuromorphic chip. We described the three typical phases; the T-phase, the N-phase, and C-phase states, described by no conspicuous dynamics, intermittent avalanche firing dynamics, and constant oscillating activities, respectively. This simple brain phase diagram was obtained by varying dynamic parameters using the Spikey chip on the firing threshold and noise level plane, followed by analyzing the power spectra. An alternative construction method was also used to obtain a similar phase diagram using the neuronal firing activities in the entire network. The phase diagram was shown to be consistent with the emulation data set and a new theory of stochastic dynamics, where noise is introduced. Our results open a new way to use an advanced neuromorphic chip for computational neuroscience studies. The application of a neuromorphic chip allows researchers to conduct experiments freely, incorporate the spatial structure of the neural network, scale up network size without exponetially increasing of emulation time, add to the understanding of neuronal dynamics with higher degrees of freedom, and guide biological experiments. Our work may pave the way for a new era of brain simulation to help accelerate and enhance research in the field of neuroscience.

## Appendix:  Materials and Methods

The emulation was done in a neuromorphic chip, Spikey, which was fabricated in a 180 nm CMOS process with die size 5 mm $×$ 5 mm. While the communication to the host computer was mostly established by digital circuits, the spiking neural network was mostly implemented with analog circuits. Compared to the biological timescale, network dynamics on the Spikey chip could be accelerated because time constants on the chip were approximately 10$4$ times shorter than in a biological system. A more detailed description can be found in section A.1.

The brain slice experiments were done at the David Geffen School of Medicine, UCLA using a standard current clamp single-neuron recording. A formal description of the experiments is in section A.2.

For comparison purposes, both emulation and brain experiment membrane potentials were normalized by their unit spike width in time. The unit spike width was defined as the full width at half maximum of a spike on the plot of membrane potential versus time. More information is in section A.3.

The brain phase diagram was constructed based on both the standard power spectrum and a correlation parameter. For computational purposes equation 4.1 was discretized and a time window of 5 ms was chosen. A detailed description is in section A.4.

### A.1  Spikey Chip and Emulation Settings

The Spikey chip was fabricated in a 180 nm CMOS process with a die size of 5 mm $×$ 5 mm. While the communication to the host computer was mostly established by digital circuits, the spiking neural network was mostly implemented with analog circuits. Compared to biological real time, network dynamics on the Spikey chip were accelerated because time constants on the chip were approximately 10$4$ times shorter than those in a biological system. Each neuron and each synapse was physically implemented on the chip. The total number of 384 neurons was split into two blocks of 192 neurons with 256 synapses each. Each line of synapses in these blocks (192 synapses) was driven by a line driver that could be configured to receive input from external spike sources (e.g., generated from a host computer) from on-chip neurons in the same block or from on-chip neurons in the adjacent block. Table 2 shows the accessible parameters of each individual neuron and each synapse in the Spikey chip.

Table 2:
Accessible Parameters for the Individual Neuron and Synapse on the Spikey Chip.
 Scope Name Type Description Neuron circuits $τrefrac$ Bias current controlling neuron refractory time $Vth$ Firing threshold voltage $Vreset$ Reset potential Synapse and line drivers $NA$ Two digital configuration bits selecting input of line driver $NA$ Two digital configuration bits setting line excitatory or inhibitory $trise,tfall$ Two bias currents for rising and falling slew rate of presynaptic voltage ramp $grisemax$ Bias current controlling maximum voltage of presynaptic voltage ramp $w$ 4-bit weight of each individual synapse
 Scope Name Type Description Neuron circuits $τrefrac$ Bias current controlling neuron refractory time $Vth$ Firing threshold voltage $Vreset$ Reset potential Synapse and line drivers $NA$ Two digital configuration bits selecting input of line driver $NA$ Two digital configuration bits setting line excitatory or inhibitory $trise,tfall$ Two bias currents for rising and falling slew rate of presynaptic voltage ramp $grisemax$ Bias current controlling maximum voltage of presynaptic voltage ramp $w$ 4-bit weight of each individual synapse

Notes: Each parameter is with corresponding model parameter names and its corresponding description. Electronic parameters that have no direct translation to model parameters are denoted $NA$ (Pfeil et al. 2013).

We constructed a recurrent network of neurons with sparse and random connections on the Spikey chip. A group of 192 neurons on the chip was used in this structure. Each neuron was configured to have a fixed number $k$ = 5 of presynaptic partners randomly drawn from all other hardware neurons. A set of eight randomly chosen neurons from the full group was stimulated by an external input modeled as a uniform-time-distributed signal. The average effect of these external inputs for triggering neuronal firing was 0. In order to do so, half of the external stimuli were excitatory and half inhibitory.

### A.2  Brain Slice Experiments

In the real brain slice experiments, neocortical sample sites were excised for in vitro electrophysiological evaluation based on abnormal neuroimaging and electrocorticography (ECoG) assessments. Cortical tissue samples were obtained from two pediatric patients (3 and 5 years of age) undergoing surgery for the treatment of pharmacoresistant epilepsy. For our study, the area with the least anatomical abnormality was used for electrophysiology. In fact, one case used for this study had no remarkable pathology and is as close to normal as one can ethically obtain. Similarly, the other case had preserved cortical lamination. However, a word of caution is warranted. Although the cortical pathology was minimal, the patients suffered from epileptic seizures; thus, the intrinsic and synaptic properties of the recorded neurons may have perturbations that could make them different from those in healthy tissue.

Sample sites (2 cm$3$) were removed microsurgically and directly placed in ice-cold artificial cerebrospinal fluid (ACSF) containing (in mM); NaCl 130, NaHCO$3$ 26, KCl 3, MgCl$2$ 5, NaH$2$PO$4$ 1.25, CaCl$2$ 1.0, glucose 10 (pH 7.2-7.4). Within 5 to 10 minutes, slices (350 $μ$m) were cut (Microslicer, DSK Model 1500E or Leica VT1000S) and placed in ACSF for at least 1 hour (in this solution CaCl$2$ was increased to 2 mM and MgCl$2$ was decreased to 2 mM). Slices were constantly oxygenated with 95% O$2$-5% CO$2$ (pH 7.2–7.4, osmolality 290–300 mOsm, at room temperature). Patch electrodes (3–6 M$Ω$) were filled with (in mM) Cs-methanesulfonate 125, NaCl 4, KCl 3, MgCl$2$ 1, MgATP 5, ethylene glycol-bis ($β$-aminoethyl ether)-N,N,N',N'-tetraacetic acid (EGTA) 9, HEPES 8, GTP 1, phosphocreatine 10 and leupeptine 0.1 (pH 7.25–7.3, osmolality 280–290 mOsm) for voltage clamp recordings or K-gluconate 140, HEPES 10 MgCl$2$ 2, CaCl$2$ 0.1, EGTA 1.1, and 2 K$2$ATP (pH 7.25–7.3, osmolality 280–290 mOsm) for current clamp recordings. The access resistance ranged from 8 to 20 M$Ω$. Liquid junction potentials (6 mV) were not corrected. Cells were initially held at $-$70 mV in voltage clamp mode. At this holding potential, passive membrane properties in slices were determined by applying a depolarizing step voltage command (10 mV) and using the membrane test function integrated in the pClamp (version 8) software (Axon Instruments, Foster City, CA). This function reported membrane capacitance (in pF), input resistance (in M$Ω$), and time constant (in ms). Then the recording was switched to current clamp mode to determine the resting membrane potential and the firing threshold using depolarized current injection. As paroxysmal discharges rarely occur spontaneously in slices from cortical tissue samples, this activity was induced by a combination of the GABA$A$ receptor antagonist bicuculline (20 $μ$M) and the K-channel blocker 4-aminopyridine (100 $μ$M). Ictal activity was recorded after blockade of GABA$B$ receptors with the antagonist phaclofen (6 $μ$M).

### A.3  Emulation and Clinical Data Comparison

Having noted that the difference between ranges of the spiking signal in the two systems was too large for effective comparison, we scaled the membrane potential of both medical and emulated data to a range of 0 to 1 mV. The process has three steps. First, the graph of membrane potential versus time was shifted vertically by subtracting the minimal membrane potential from the original data for all three phases. Second, the data cursor tool was used to determine the amplitude $Δ$A of subtracted membrane potential of the seizure phase. And third, the membrane potential signal for the three phases was rescaled by dividing the subtracted membrane potential with $Δ$A.

Due to the intrinsic spiking frequency difference, a need to normalize the data arose prior to the comparison between the clinical and emulated data. In order to compare the shape of power spectra for clinical data and emulated data, the time-series membrane potential recordings, whose signal range has been scaled to 0 $∼$ 1 mV, were normalized with a unit spike width. The unit spike width was $Δt1$ for medical data and $Δt2$ for emulated data. As shown in Figure 6 (top), the unit spike width was defined as the full width at half maximum of a spike on the graph of the membrane potential versus time. The time coordinates of the end points were determined using the data cursor tool in Matlab.

Figure 6:

Data normalization and order parameter extraction. (Top) The full width at half maximum was used to define the unit spike time for two different sets of data. (Top left) $Δt1$ for clinical data. (Top right) $Δt2$ for emulated data. (Bottom) Schematics of how to use the raster plot of neurons to extract the correlation parameter. Each short vertical line represents the occurrence of neuron spiking. The row vectors corresponding to the raster plot of neuron $i$ and $j$ are [0,1,0,1,1] and [1,0,0,1,0], respectively.

Figure 6:

Data normalization and order parameter extraction. (Top) The full width at half maximum was used to define the unit spike time for two different sets of data. (Top left) $Δt1$ for clinical data. (Top right) $Δt2$ for emulated data. (Bottom) Schematics of how to use the raster plot of neurons to extract the correlation parameter. Each short vertical line represents the occurrence of neuron spiking. The row vectors corresponding to the raster plot of neuron $i$ and $j$ are [0,1,0,1,1] and [1,0,0,1,0], respectively.

Standard power spectrum analysis was then applied to the normalized time-series membrane potential data.

### A.4  Correlation Parameter Extraction

In order to fully accertain and articulate our proposed picture, we used an alternative approach to construct the brain phase diagram. The literature has shown that it might also be advantageous to quantify brain dynamics by defining a correlation parameter $〈Csyn(Vth)〉$ as the average over the correlations between pairs of neurons i,j. For computational purposes, we discreted the equation as follows:
$Ci,j(τ)=1τ+1∑t=t0t0+τσi(t)σj(t),$
(A.1)
where $σi(t)$ is 1 if the neuron i spiked at time t, and zero otherwise. This quantity was evaluated over a time window $τ$ 5 ms, which was approximately the typical stimulation time period of the chip.

Since it is difficult to observe the spiking of neurons at a fixed discrete time value in reality, we divided the time window $τ$ into five time bins. The parameter was extracted from the raster plot of spike train. Each point $(i,t)$ on the raster plot represents the spiking activity of neuron $i$ at time bin $t$. To extract this parameter, a matrix representation of this raster plot was constructed using Matlab. In our experiment, there were 192 neurons, and each neuron was observed over 1000 ms. Hence, we constructed a 192 $×$ 1000 zero-one matrix $Σi,t$ to represent each neuron's spiking activity as shown in the raster plot. Each row corresponded to a neuron, whereas each column corresponded to a time bin. If the neuron $i$ spikes at time bin $t$, we denote $Σit=1$.

The parameter $Ci,j$ for each pair of neurons i,j can then be extracted from the matrix $Σi,t$ in three steps: (1) perform elementwise multiplication on the row vectors $Σi$ and $Σj$ to obtain a resultant vector, denoted as $mij$, (2) sum up the entries in $mij$ and divide by a constant $(τ+1)$ as indicated in the equation, and (3) divide the result obtained in the previous step by 200, given 200 windows, to obtain the average correlations at certain conditions (threshold noise level). Finally, given a total of 192 neurons, there were 18,336 pairwise correlations $Ci,j$, $1≤i≠j≤192$, at each threshold level. The average over all the correlations $Ci,j$ was the value of $〈Csyn(Vth)〉$ at threshold level $Vth$. Table 3 summarizes the typical values for $〈Csyn〉$ in the different phases.

Table 3:
Typical $〈Csync〉$ under Different Phases.
 Phase Typical $〈Csync〉$ T-phase $<10-6$ N-phase $∼10-4$ C-phase $>10-3$
 Phase Typical $〈Csync〉$ T-phase $<10-6$ N-phase $∼10-4$ C-phase $>10-3$

## Acknowledgments

K.L.W. acknowledges the support of the endowed Raytheon professorship. This work is in part supported by ARO under W911NF-15-1-0561:P00001. The neuromorphic hardware and software is partially supported by EU Grant 269921 (BrainScaleS) and EU Grant 604102 (Human Brain Project, HBP). The NIH grant U54HD087101-01 supports the Cell, Circuits and Systems Analysis Core. We thank Thomas Pfeil for his support of the hardware system.

## References

Beggs
,
J. M.
(
2008
).
The criticality hypothesis: How local cortical networks might optimize information processing
.
Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences
,
366
(
1864
),
329
343
.
Beggs
,
J. M.
, &
Plenz
,
D.
(
2003
).
Neuronal avalanches in neocortical circuits
.
Journal of Neuroscience
,
23
(
35
),
11167
11177
.
Beggs
,
J. M.
, &
Plenz
,
D.
(
2004
).
Neuronal avalanches are diverse and precise activity patterns that are stable for many hours in cortical slice cultures
.
Journal of Neuroscience
,
24
(
22
),
5216
5229
.
Beggs
,
J. M.
, &
Timme
,
N.
(
2012
).
Being critical of criticality in the brain
.
Frontiers in Physiology
,
3
,
163
.
Benayoun
,
M.
,
Cowan
,
J. D.
,
van Drongelen
,
W.
, &
Wallace
,
E.
(
2010
).
Avalanches in a stochastic model of spiking neurons
.
PLoS Computational Biology
,
6
(
7
),
e1000846
.
Bonachela
,
J. A.
,
De Franciscis
,
S.
,
Torres
,
J. J.
, &
Mũnoz
,
M. A.
(
2010
).
Self-organization without conservation: Are neuronal avalanches generically critical
?
Journal of Statistical Mechanics: Theory and Experiment
,
2010
(
2
),
P02015
.
Bornholdt
,
S.
, &
Rohlf
,
T.
(
2000
).
Topological evolution of dynamical networks: Global criticality from local dynamics
.
Physical Review Letters
,
84
(
26
),
6114
.
Bornholdt
,
S.
, &
Röhl
,
T.
(
2003
).
Self-organized critical neural networks
.
Physical Review Letters
,
67
(
6
),
066118
.
Brunel
,
N.
(
2000
).
Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons
.
Journal of Computational Neuroscience
,
8
(
3
),
183
208
.
Chialvo
,
D. R.
(
2010
).
Emergent complex neural dynamics
.
Nature Physics
,
6
(
10
),
744
750
.
Garrity
,
A. G.
,
Pearlson
,
G. D.
,
McKiernan
,
K.
,
Lloyd
,
D.
,
Kiehl
,
K. A.
, &
Calhoun
,
V. D.
(
2007
).
Aberrant default mode functional connectivity in schizophrenia
.
American Journal of Psychiatry
,
164
(
3
),
450
457
.
Gireesh
,
E. D.
, &
Plenz
,
D.
(
2008
).
Neuronal avalanches organize as nested theta and beta/gamma-oscillations during development of cortical layer 2/3
.
Proceedings of the National Academy of Sciences
,
105
(
21
),
7576
7581
.
Hahn
,
G.
,
Petermann
,
T.
,
Havenith
,
M. N.
,
Yu
,
S.
,
Singer
,
W.
,
Plenz
,
D.
, &
Nikoli
,
D.
(
2010
).
Neuronal avalanches in spontaneous activity in vivo
.
Journal of Neurophysiology
,
104
(
6
),
3312
3322
.
Haimovici
,
A.
,
Tagliazucchi
,
E.
,
Balenzuela
,
P.
, &
Chialvo
,
D. R.
(
2013
).
Brain organization into resting state networks emerges at criticality on a model of the human connectome
.
Physical Review Letters
,
110
(
17
),
178101
.
Hartley
,
C.
,
Taylor
,
T. J.
,
Kiss
,
I. Z.
,
Farmer
,
S. F.
, &
Berthouze
,
L.
(
2014
).
Identification of criticality in neuronal avalanches: II. A theoretical and empirical investigation of the driven case
.
Journal of Mathematical Neuroscience
,
4
(
1
),
9
.
He
,
Y.
,
Chen
,
Z.
, &
Evans
,
A.
(
2008
).
Structural insights into aberrant topological patterns of large-scale cortical networks in Alzheimer's disease
.
Journal of Neuroscience
,
28
(
18
),
4756
4766
.
Hesse
,
J.
, &
Gross
,
T.
(
2014
).
Self-organized criticality as a fundamental property of neural systems
.
Frontiers in Systems Neuroscience
,
8
,
166
.
Hobbs
,
J. P.
,
Smith
,
J. L.
, &
Beggs
,
J. M.
(
2010
).
Aberrant neuronal avalanches in cortical tissue removed from juvenile epilepsy patients
.
Journal of Clinical Neurophysiology
,
27
(
6
),
380
386
.
Hudson
,
A. E.
,
Calderon
,
D. P.
,
Pfaff
,
D. W.
, &
Proekt
,
A.
(
2014
).
Recovery of consciousness is mediated by a network of discrete metastable activity states
.
Proceedings of the National Academy of Sciences
,
111
(
25
),
9283
9288
.
Juanico
,
D. E.
, &
Monterola
,
C.
(
2007
).
Background activity drives criticality of neuronal avalanches
.
Journal of Physics A: Mathematical and Theoretical
,
40
(
31
),
9297
.
Jung
,
P.
,
Cornell-Bell
,
A.
,
,
K. S.
, &
Moss
,
F.
(
1998
).
Noise-induced spiral waves in astrocyte syncytia show evidence of self-organized criticality
.
Journal of neurophysiology
,
79
(
2
),
1098
1101
.
Kello
,
C. T.
,
Brown
,
G. D.
,
Ferrer-i-Cancho
,
R.
,
Holden
,
J. G.
,
,
K.
,
Rhodes
,
T.
, &
Van Orden
,
G. C.
(
2010
)
Scaling laws in cognitive sciences
.
Trends in Cognitive Sciences
,
14
(
5
),
223
232
.
Laufs
,
H.
,
Hamandi
,
K.
,
,
A.
,
Kleinschmidt
,
A. K.
,
Duncan
,
J. S.
, &
Lemieux
,
L.
(
2007
).
Temporal lobe interictal epileptic discharges affect cerebral activity in default mode brain regions
.
Human Brain Mapping
,
28
(
10
),
1023
1032
.
Lee
,
K. E.
,
Lopes
,
M. A.
,
Mendes
,
J. F. F.
, &
Goltsev
,
A. V.
(
2014
).
Critical phenomena and noise-induced phase transitions in neuronal networks
.
Physical Review E
,
89
(
1
),
012701
.
Marković
,
D.
, &
Gros
,
C.
(
2014
).
Power laws and self-organized criticality in theory and nature
.
Physics Reports
,
536
(
2
),
41
74
.
Meisel
,
C.
, &
Gross
,
T.
(
2009
).
Adaptive self-organization in a realistic neural network model
.
Physical Review E
,
80
(
6
),
061917
.
Meisel
,
C.
,
Olbrich
,
E.
,
Shriki
,
O.
, &
Achermann
,
P.
(
2013
).
Fading signatures of critical brain dynamics during sustained wakefulness in humans
.
Journal of Neuroscience
,
33
(
44
),
17363
17372
.
Meisel
,
C.
,
Storch
,
A.
,
Hallmeyer-Elgner
,
S.
,
Bullmore
,
E.
, &
Gross
,
T.
(
2012
).
Failure of adaptive self-organized criticality during epileptic seizure attacks
.
PLoS Comput Biol.
,
8
(
1
),
e1002312
.
Moosavi
,
S. A.
, &
Montakhab
,
A.
(
2014
).
Mean-field behavior as a result of noisy local dynamics in self-organized criticality: Neuroscience implications
.
Physical Review E
,
89
(
5
),
052139
.
Moretti
,
P.
, &
Muñoz
,
M. A.
(
2013
).
Griffiths phases and the stretching of criticality in brain networks
.
Nature Communications
,
4
,
2521
.
Osorio
,
I.
,
Frei
,
M. G.
,
Sornette
,
D.
,
Milton
,
J.
, &
Lai
,
Y. C.
(
2010
).
Epileptic seizures: Quakes of the brain
?
Physical Review E
,
82
(
2
),
021919
.
Ovchinnikov
,
I. V.
(
2016
).
Introduction to supersymmetric theory of stochastics
.
Entropy
,
18
(
4
),
108
.
Ovchinnikov
,
I. V.
,
Li
,
W.
,
Sun
,
Y.
,
Schwartz
,
R. N.
,
Hudson
,
A. E.
,
Meier
,
K.
, &
Wang
,
K. L.
(
2016
)
Criticality or supersymmetry breaking
?
arXiv:1609.00001
.
Ovchinnikov
,
I. V.
, &
Wang
,
K. L.
(
2017
).
Stochastic dynamics and combinatorial optimization
.
Modern Physics Letters B
,
31
(
31
),
1750285
.
Palva
,
J. M.
,
Zhigalov
,
A.
,
Hirvonen
,
J.
,
Korhonen
,
O.
,
,
K.
, &
Palva
,
S.
(
2013
).
Neuronal long-range temporal correlations and avalanche dynamics are correlated with behavioral scaling laws
.
Proceedings of the National Academy of Sciences
,
110
(
9
),
3585
3590
.
Petermann
,
T.
,
Thiagarajan
,
T. C.
,
Lebedev
,
M. A.
,
Nicolelis
,
M. A.
,
Chialvo
,
D. R.
, &
Plenz
,
D.
(
2009
).
Spontaneous cortical activity in awake monkeys composed of neuronal avalanches
.
Proceedings of the National Academy of Sciences
,
106
(
37
),
15921
15926
.
Pfeil
,
T.
,
Grübl
,
A.
,
Jeltsch
,
S.
,
Müller
,
E.
,
Müller
,
P.
,
Petrovici
,
M. A.
, …
Meier
,
K.
(
2013
).
Six networks on a universal neuromorphic computing substrate
.
Frontiers in Neuroscience
,
7
,
11
.
Plenz
,
D.
, &
Thiagarajan
,
T. C.
(
2007
).
The organizing principles of neuronal avalanches: Cell assemblies in the cortex?
Trends in Neurosciences
,
30
(
3
),
101
110
.
Poil
,
S. S.
,
Hardstone
,
R.
,
Mansvelder
,
H. D.
, &
,
K.
(
2012
).
Critical-state dynamics of avalanches and oscillations jointly emerge from balanced excitation/inhibition in neuronal networks
.
Journal of Neuroscience
,
32
(
29
),
9817
9823
.
Priesemann
,
V.
,
Wibral
,
M.
,
Valderrama
,
M.
,
Pröpper
,
R.
,
Le Van Quyen
,
M.
,
Geisel
,
T.
,
Triesch
,
J.
, …
Munk
,
M. H.
(
2014
).
Spike avalanches in vivo suggest a driven, slightly subcritical brain state
.
Frontiers in Systems Neuroscience
,
8
,
108
.
Shew
,
W. L.
, &
Plenz
,
D.
(
2013
).
The functional benefits of criticality in the cortex
.
Neuroscientist
,
19
(
1
),
88
100
.
Shriki
,
O.
,
Alstott
,
J.
,
Carver
,
F.
,
Holroyd
,
T.
,
Henson
,
R. N.
,
Smith
,
M. L.
,
Coppola
,
R.
, …
Plenz
,
D.
(
2013
).
Neuronal avalanches in the resting meg of the human brain
.
Journal of Neuroscience
,
33
(
16
),
7079
7090
.
Tagliazucchi
,
E.
,
Balenzuela
,
P.
,
Fraiman
,
D.
, &
Chialvo
,
D. R.
(
2012
).
Criticality in large-scale brain fmri dynamics unveiled by a novel point process analysis
.
Frontiers in Physiology
,
3
,
15
.
Taylor
,
T. J.
,
Hartley
,
C.
,
Simon
,
P. L.
,
Kiss
,
I. Z.
, &
Berthouze
,
L.
(
2013
).
Identification of criticality in neuronal avalanches: I. A theoretical investigation of the non-driven case
.
Journal of Mathematical Neuroscience
,
3
(
1
),
5
.
Torres
,
J. J.
, &
Marro
,
J.
(
2015
).
Brain performance versus phase transitions
.
Scientific Reports
,
5
,
12216
.
Williams-Garca
,
R. V.
,
Moore
,
M.
,
Beggs
,
J. M.
, &
Ortiz
,
G.
(
2014
).
Quasicritical brain dynamics on a nonequilibrium widom line
.
Physical Review E
,
90
(
6
),
062714
.
Yu
,
S.
,
Yang
,
H.
,
Nakahara
,
H.
,
Santos
,
G. S.
,
Nikolić
,
D.
, &
Plenz
,
D.
(
2011
)
Higher-order interactions characterized in cortical activity
.
Journal of Neuroscience
,
31
(
48
),
17514
17526
.

## Notes

1

We want to point out that there is a discrepancy in the terminology to describe the neuronal activity threshold between the neuromorphic and neuroscience communities (for example in Figures 4 and 5). The neuromorphic community uses ''increased membrane potential'' to describe neuron hyperpolarization when the membrane potential changes from −70 mV to around −55 mV. Whereas neuroscientists use ''decreased membrane potential.'' As the numbers are negative, the closer the membrane is to −55 mV, the more likely the neuron fires and the farther away (more negative, −70 mV) it is less likely to fire action potentials. Here we use the former terminology throughout the paper to avoid ambiguity.