Critical-like Brain Dynamics in a Continuum from Second- to First-Order Phase Transition

Sheng H. Wang; Felix Siebenhühner; Gabriele Arnulfo; Vladislav Myrov; Lino Nobili; Michael Breakspear; Satu Palva; J. Matias Palva

doi:10.1523/JNEUROSCI.1889-22.2023

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Figure 1.
Bistability is caused by elevated positive local feedback. A, The order parameter R(t) increases (top) as phases (Ph.) of the oscillators in the model become increasingly synchronized (bottom). Re/Im, Real/imaginary part of the complex R. B, Exemplary segments of order R time series when the model is in the sub-, super-, critical, and bistable phases indicated in F. C–E, Criticality estimates as a function of the local positive feedback (ρ) and neuronal coupling (κ). C, The mean order. Two arrows indicate the amount of coupling required for the model to transition from asynchrony (R = 0.1) to hypersynchrony (R = 0.9). D, The DFA exponent as a measure of LRTCs in the fluctuations of R(t). E, The BiS assessed from the R² time series. C–E, Each pixel is the mean of 50 simulations. F, An overlaid regimen map based on observation from C–E; classic criticality (with a second-order phase transition) is associated with small positive feedback ρ (black dashed); bistable criticality is seen at mid-to-high degree of ρ (enclave inside red line). G, Probability distribution of R in bistable (top) and classic (bottom) criticality. The peak DFA (black line) coincided with the phase transition (i.e., moderate R). H, Probability density (pdf) of R and (I) the detrended fluctuations (DF) and the DFA exponents (the slope) of the time series from B, color-coded.
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Figure 2.
Bistability and LRTCs were robust, large-scale phenomena in the resting-state brain. A, B, Five minutes of broad-band and narrow-band power (R²) time series from (A) a MEG parcel in visual area (Vis) in 1 subject and (B) an SEEG contact in middle frontal gyrus in one patient. Insets, Bistability as narrow-band traces switching between “up” and “down” states. C, Group-level probability (z axis) distribution of narrow-band (y axis) mean amplitude (R). D, DFA exponents. E, BiS estimates. Data were pooled over all nEZ SEEG contacts or MEG parcels; subject and contact/parcel number indicated in C. Black lines indicate mean of real data. Red dashed lines indicate 99th percentile of surrogate observation. F–H, Examples of narrow-band DFA and BiS probability distribution as indicated by colored arrows in D, E.
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Figure 3.
Bistability and LRTCs were coexisting, correlated phenomena in resting-state MEG and SEEG. Neuroanatomical similarity (Spearman's correlation) between group-average narrow-band BiS and DFA estimates of (A) MEG and (B) SEEG in Schaefer 100-parcel atlas. Red boxes represent frequency clusters showing high similarity. C, Narrow-band group-averaged estimates were collapsed into θ−α (5.4–11 Hz) and γ (40–225 Hz) band based on similarity shown in A. White-out columns in SEEG data represent excluded parcels because of insufficient sampling. D, E, Parcel-wise group-average θ−α band BiS maps for (D) MEG and (E) SEEG. F, Kruskal–Wallis one-way ANOVA for group-level differences in DFA and BiS estimates between Yeo systems. Dashed line indicates –log₁₀(p value) > 1.3 (i.e., p < 0.05). Correlations between group-average parcel BiS and DFA estimates in θ−α (cross) and γ band (circles) in (G) MEG and (H) SEEG, −log10(p) > 5 (FDR-corrected). I, Spearman's correlations between within-subject-average BiS and DFA estimates in Yeo systems (subject N_MEG per system =18; N_SEEG per system = 50 ± 9.4, range: 36–60, variable SEEG subject N per system because of heterogamous spatial sampling).
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Figure 4.
Executive functions were correlated with θ−α band DFA and BiS estimates in MEG subjects. A, Spearman's correlation between subject neuropsychological test scores and within subject mean parcel θ−α band DFA. B, BiS estimates collapsed over parcels. Dashed lines indicate 5th and 95th percentile of correlations for surrogate data (N_surrogate = 10⁵, FDR-corrected). C, D, Subject Zoo map time rank and θ−α band parcel-collapsed (C) DFA and (D) BiS estimates. Each marker represents 1 subject. E, Fraction of parcels that showed significant correlation between neuropsychological test scores and individual parcel θ−α band DFA and (G) BiS estimates (p < 0.05, FDR-corrected). F, Parcels showing significant correlations between Zoom map time scores and θ−α band DFA and (H) BiS estimates.
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Figure 5.
Bistability showed strong predictive power for epileptogenic pathophysiology. A, B, Five minutes of broad-band traces and narrow-band power (R²) time series of an EZ (A) and an nEZ (B) cortical location recorded with two distinct electrode shafts in 1 subject. The two contacts were 19.7 mm apart within the supervisor frontal gyrus, and they were referenced with the same nearest white matter contact (Arnulfo et al., 2015a). C, Average normalized narrow-band BiS and (D) DFA estimates for all EZ (pink) and nEZ contacts (green) of the patient cohort. Shades represent 25th and 75th percentiles. E, The effect size of differences between EZ and nEZ contacts in frequency-collapsed BiS (red) and DFA (black). Dashed line indicates 99th percentile observation from surrogate data (N_surrogate =1000). F, Feature importance estimated using SHAP values. G, The AUC of receiver operating characteristics averaged across subjects (black) and the AUC of pooled within-subject classification results (blue) when using (1) DFA alone, (2) BiS alone, (3) D&B, and (4) D&B plus contact loci in Yeo systems (D&B(Y)). Dashed lines indicate 99th percentile of AUC observed from 1000 surrogates created independently for each of the four feature sets. H–J, Post hoc inspection of results derived using D&B(Y) feature set (black marker in G). H, Spearman's correlation (p < 10⁻⁶, n = 55) between individual AUC and within-subject mean Cohen's d between EZ and nEZ in band-collapsed DFA and BiS. I, Receiver operating characteristics of classification within subjects (thin lines) and mean receiver operating characteristic (thick). J, Within-patient prediction precision as a function of TPR indicated by the magenta box from I. Red marker represents the population mean. Precision = true positive ÷ reported positive.

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Table 1.

Demographic data for the SEEG patient cohort^{^a}

ID	EZ location	Age (yr)	Sex	Medication	Outcome (Engel score)
1	Right mesial frontal	21	M	Carbamazepine 600 mg, Levetiracetam 1500 mg	IB (8 yr)
2	Right temporal insular	26	M	Carbamazepine 1200 mg, Primidone 750 mg, Clonazepam 10 mg	IA (37 mo)
3	Right temporal	38	M	Carbamazepine 1200 mg, Dilantin 450 mg, Lacosamide 300 mg, Clonazepam 10 mg	No surgery
4	Left temporo-parietal	38	F	Phenobarbital 100 mg, Topiramate 100 mg, Levetiracetam 3000 mg	IA (25 mo)
5	Left temporal-insular	24	M	Oxcarbazepine 600 mg, Lacosamide 400 mg	IA (15 mo)
6	Right temporo-insular	40	F	Levetiracetam 1000 mg, Lacosamide 350 mg, Sertraline 50 mg, Lorazepam 1 mg	IIIA (32 mo)
7	Precuneus	19	M	Oxcarbazepine 600 mg, Lacosamide 400 mg	No surgery
8	Functional epilepsy	20	F	Carbamazepine 1200 mg, Levetiracetam 1500 mg	No surgery
9	Right occipito-temporo-parietal	20	M	Lamotrigine 400 mg, Levetiracetam 3000 mg, Lacosamide 400 mg	IA (3 yr)
10	Left temporal	35	M	Topiramate 200 mg, Carbamazepine 900 mg	IA (12 mo)
11	Right temporal anterior	28	M	Carbamazepine 1200 mg	IIA (66 mo)
12	Temporo-hip	36	F	Carbamazepine 1400 mg, Levetiracetam 3000 mg	IIA (6 mo)
13	Left frontal anterior	40	M	Levetiracetam 2750 mg, Carbamazepine 800 mg, Primidone 750 mg	IA (12 mo)
14	Thermo-coagulation multiple sites	39	M	Oxcarbazepine 1800 mg, Clobazam 20 mg	IA (36 mo)
15	Right fronto-temporo-insular	24	F	Carbamazepine 1000 mg, Clobazam 20 mg, Lamotrigine 200 mg	IVA (24 mo)
16	Left parieto-opercolo-insular	31	F	Carbamazepine 1200 mg, Clobazam 40 mg, Phenobarbital 75 mg	IVA (12 mo)
17	Right temporo-perisylvian	34	F	Phenobarbital 150 mg, Lacosamide 400 mg, Clobazam 10 mg	IIC (38 mo)
18	Right perisylvian-insular	17	M	Carbamazepine 800 mg, Lamotrigine 400 mg	IVA (26 mo)
19	Right temporo-parieto-occipital	36	M	Oxcarbazepine 1200 mg, Phenobarbital 150 mg, Valproate 1000 mg	IA (35 mo)
20	—	32	F	Carbamazepine 700 mg	No surgery
21	Left temporal antero-mesial	32	M	Carbamazepine 1200 mg, Levetiracetam 750 mg	IA (61 mo)
22	Right fronto-centro-insular	33	M	Carbamazepine 800 mg, Lacosamide 800 mg, Zonisamide 250 mg	IIA (38 mo)
23	Left temporal	21	F	Levetiracetam 1750 mg, Lacosamide 400 mg, Valproate1000 mg	IA (24 mo)
24	Right parietal	23	M	Levetiracetam 3000 mg, CBZ 1000 mg, Lacosamide 500 mg	IA (24 mo)
25	Thermo-coagulation	46	M	Carbamazepine 1200 mg, Phenobarbital 100 mg	IA (24 mo)
26	Right frontal	20	F	Valproate 800 mg, Clobazam 10 mg	IIA (36 mo)
27	Right fronto-mesial	21	M	Carbamazepine 800 mg, Levetiracetam 3000 mg, Nitrazepam 1.5 mg	IIIA (13 mo)
28	Right fronto-central	22	M	Lamotrigine 400 mg, Levetiracetam 2000 mg	IA (24 mo)
29	Right frontal	20	M	Carbamazepine 600 mg, Rufinamide 1500 mg	IVA (13 mo)
30	Right frontal	44	F	Carbamazepine 1200 mg, Zonisamide 400 mg, Phenobarbital 1000 mg	IC (24 mo)
31	—	17	M	Carbamazepine 300 mg	No surgery
32	—	14	M	Levetiracetam 1500 mg, Clobazam 5 mg	No surgery
33	Right temporal antero-mesial	30	F	Oxcarbazepine 2000 mg, Phenobarbital 150 mg	IIA (36 mo)
34	—	24	M	Carbamazepine 16000 mg, Levetiracetam 4000 mg	No surgery
35	—	29	F	Levetiracetam 3000 mg	No surgery
36	Right orbito-temporal	29	F	Zonisamide 400 mg, Levetiracetam 750 mg, Carbamazepine 1400 mg	IA (62 mo)
37	—	45	F	Lacosamide 500 mg, Valproate 1000 mg, Zonisamide 200 mg	No surgery
38	Thermo-coagulation multiple sites	34	F	Carbamazepine 1000 mg, Levetiracetam 2500 mg	IA (12 mo)
39	Thermo-coagulation multiple sites	50	M	Levetiracetam 2000 mg, Lacosamide 600 mg	IIA (6 mo)
40	Left occipital	17	F	Carbamazepine 1200 mg, Levetiracetam 1500 mg, Lacosamide 300 mg	IB (49 mo)
41	Right temporal	44	F	Topiramate 300 mg, Oxcarbazepine 1200 mg	IIA (50 mo)
42	—	27	M	Carbamazepine 800 mg, Lamotrigine 200 mg	No surgery
43	—	46	M	Carbamazepine 1200 mg, Levetiracetam 3000 mg, Lacosamide 150 mg, Clobazam 20 mg	No surgery
44	Right antero-frontal	28	M	Carbamazepine 1000 mg, Levetiracetam 1000 mg	IIIA (61 mo)
45	Thermo-coagulation right temporo-parieto-perisylvian	27	F	Topiramate 200 mg, Lamotrigine 200 mg	IA (5 yr)
46	Right temporal antero-mesial	42	F	Lacosamide 500 mg	IB (36 mo)
47	Left parieto-temporal	15	M	Carbamazepine 900 mg	IA (6 mo)
48	Thermo-coagulation right temporo-opercular	37	M	Carbamazepine 900 mg, Levetiracetam 3000 mg	IVA (12 mo)
49	Left frontal	30	F	Carbamazepine 1200 mg, Lamotrigine 200 mg, Clobazam 20 mg	IA (5 yr)
50	Left frontal	15	F	Levetiracetam 1250 mg, Oxcarbazepine 1200 mg	IA (4 yr)
51	Thermo-coagulation	41	M	Levetiracetam 3000 mg, Lacosamide 400 mg	IA (2 yr)
52	Right temporo-occipital	37	M	Lamotrigine 600 mg, Levetiracetam 2000 mg	IA (2 yr)
53	Right temporal	29	M	Carbamazepine 1400 mg, Levetiracetam 3000 mg, Clobazam 10 mg	IA (31 mo)
54	Left opercolo-insular	10	F	Carbamazepine 800 mg	IIIA (34 mo)
55	Right temporo-frontal	40	F	Lamotrigine 600 mg, Clobazam 20 mg, Phenytoin 500 mg	IA (13 mo)
56	Left temporo-insular-operculum	29	F	Carbamazepine 1800 mg, Clobazam 20 mg	IA (24 mo)
57	Right temporal	27	M	Lamotrigine 400 mg, Topiramate 400 mg	IA (24 mo)
58	—	26	M	Phenytoin 400 mg, Topiramate 500 mg	No surgery
59	Left temporal	17	M	Oxcarbazepine 1500 mg, Clobazam 20 mg, Levetiracetam 2500 mg	IA (36 mo)
60	Right temporo-mesial	25	F	Topiramate 75 mg, Carbamazepine 1500 mg	IA (12 mo)
61	Nodular heterotopia	24	F	Carbamazepine 1000 mg, Levetiracetam 500 mg, Clobazam 20 mg	IVA (12 mo)
62	Left temporo-perisylvian	37	F	Carbamazepine 1200 mg, Lamotrigine 550 mg	IIIA (12 mo)
63	Left temporal antero-mesial	32	F	Clobazam 20 mg, Phenobarbital 45 mg	IIA (55 mo)
64	Right temporo-occipital	44	M	Carbamazepine 800 mg, Levetiracetam 3000 mg, Phenobarbital 125 mg	IA (24 mo)

↵^aEZ location indicates the supposed brain location of the EZ. Thermo-coagulation indicates the practice used in surgical intervention of drug-resistant focal epilepsy where current is injected in a bipolar derivation in order to increase the pericontact temperature. —, no single focal location was identified. Age indicates the age of the patients at the recording date. Drugs are reported with their active principal names. Drug dosage is expressed in milligrams and refers to the morning dosage measured at the day of the recording. Outcome is expressed as Engel scores. Numbers in parentheses indicate the point time after surgery when the visit occurred.