Brain activity:

• interictal state
  normal (stochastic/quasi-deterministic) activity

Brain activity:

• interictal state
  normal (stochastic/quasi-deterministic) activity
• ictal state
  abnormal synchronous activity

• Data: Raw multichannel EEG/MEG/iEEG recording during ictal and interictal states.

• Data: Raw multichannel EEG/MEG/iEEG recording during ictal and interictal states.
• Goal: Detect the epileptic seizure.

• Data: Raw multichannel EEG/MEG/iEEG recording during ictal and interictal states.
• Goal: Detect the epileptic seizure.

• Data: Raw multichannel EEG/MEG/iEEG recording during ictal and interictal states.
• Goal: Detect the epileptic seizure.

• Automated detection.
  $~~~~$Currently, diagnosis relies on visual inspection of physiological recordings by the doctor, added to clinical symptoms.

• Automated detection.
  $~~~~$Currently, diagnosis relies on visual inspection of physiological recordings by the doctor, added to clinical symptoms.


• Universality.
  $~~~~$The data may correspond to different methods of recording (MEG, EEG, iEEG), numbers of sensors, types of epilepsy and particular clinical conditions of the patients.

• Automated detection.
  $~~~~$Currently, diagnosis relies on visual inspection of physiological recordings by the doctor, added to clinical symptoms.


• Universality.
  $~~~~$The data may correspond to different methods of recording (MEG, EEG, iEEG), numbers of sensors, types of epilepsy and particular clinical conditions of the patients.


• Robustness.
  $~~~~$The dataset present noise and it is sensitive to other neural conditions.

• Automated detection.
  $~~~~$Currently, diagnosis relies on visual inspection of physiological recordings by the doctor, added to clinical symptoms.


• Universality.
  $~~~~$The data may correspond to different methods of recording (MEG, EEG, iEEG), numbers of sensors, types of epilepsy and particular clinical conditions of the patients.


• Robustness.
  $~~~~$The dataset present noise and it is sensitive to other neural conditions.


• Real-time analysis.
  $~~~~$Methods of classification of ictal/interictal states rely on preprocessing of the data, training the algorithms and computational resources.

• Amplitude. Differences between the variable's extreme values.


• Short-Time Fourier Transform (STFT), an estimate of the short-term time-localized frequency content.


• Lempel-Ziv complexity (LZ), measures diversity of the patterns present in a particular signal.


• Permutation entropy, a robust non-linear statistical measure to quantify the complexity of a dynamical system in terms of a time series.

• Let $\varphi_1, \varphi_2, \dots, \varphi_n:[0,T]\to \mathbb{R}$ be a collection of real-valued time series, such as multichannel EEG/iEEG/MEG recordings.

• Let $\varphi_1, \varphi_2, \dots, \varphi_n:[0,T]\to \mathbb{R}$ be a collection of real-valued time series, such as multichannel EEG/iEEG/MEG recordings.


• We assume there exist an underlying dynamical system $(X, \phi)$ such that the time series $\{\varphi_i\}_{1\leq i \leq n}$ are the result of observation functions $\{F_i:X\to \mathbb R\}_{1\leq i \leq n}$
  (i.e. $\varphi_i(t) = F_i(\phi(t,x))$ for some initial state $x\in X$).

• Let $\varphi_1, \varphi_2, \dots, \varphi_n:[0,T]\to \mathbb{R}$ be a collection of real-valued time series, such as multichannel EEG/iEEG/MEG recordings.


• We assume there exist an underlying dynamical system $(X, \phi)$ such that the time series $\{\varphi_i\}_{1\leq i \leq n}$ are the result of observation functions $\{F_i:X\to \mathbb R\}_{1\leq i \leq n}$
  (i.e. $\varphi_i(t) = F_i(\phi(t,x))$ for some initial state $x\in X$).


• We analyze the underlying (unknown) dynamics by studying their attractors (that is, the asymptotic states of the system).

• Let $\varphi_1, \varphi_2, \dots, \varphi_n:[0,T]\to \mathbb{R}$ be a collection of real-valued time series, such as multichannel EEG/iEEG/MEG recordings.


• We assume there exist an underlying dynamical system $(X, \phi)$ such that the time series $\{\varphi_i\}_{1\leq i \leq n}$ are the result of observation functions $\{F_i:X\to \mathbb R\}_{1\leq i \leq n}$
  (i.e. $\varphi_i(t) = F_i(\phi(t,x))$ for some initial state $x\in X$).


• We analyze the underlying (unknown) dynamics by studying their attractors (that is, the asymptotic states of the system).

• Let $\varphi_1, \varphi_2, \dots, \varphi_n:[0,T]\to \mathbb{R}$ be a collection of real-valued time series, such as multichannel EEG/iEEG/MEG recordings.


• We assume there exist an underlying dynamical system $(X, \phi)$ such that the time series $\{\varphi_i\}_{1\leq i \leq n}$ are the result of observation functions $\{F_i:X\to \mathbb R\}_{1\leq i \leq n}$
  (i.e. $\varphi_i(t) = F_i(\phi(t,x))$ for some initial state $x\in X$).


• We analyze the underlying (unknown) dynamics by studying their attractors (that is, the asymptotic states of the system).

• Let $\varphi_1, \varphi_2, \dots, \varphi_n:[0,T]\to \mathbb{R}$ be a collection of real-valued time series, such as multichannel EEG/iEEG/MEG recordings.


• We assume there exist an underlying dynamical system $(X, \phi)$ such that the time series $\{\varphi_i\}_{1\leq i \leq n}$ are the result of observation functions $\{F_i:X\to \mathbb R\}_{1\leq i \leq n}$
  (i.e. $\varphi_i(t) = F_i(\phi(t,x))$ for some initial state $x\in X$).


• We analyze the underlying (unknown) dynamics by studying their attractors (that is, the asymptotic states of the system).

We will detect the change of dynamics via the topological analysis of embeddings of the signals.

• Let $\varphi = (\varphi_1, \varphi_2, \dots \varphi_n):[0,T]\to \mathbb{R}^n$ be a multichannel recording.$~~~~~~~~~~~~~~~~~~$$~~~~~~~~~~~~~~~~$

• Let $\varphi = (\varphi_1, \varphi_2, \dots \varphi_n):[0,T]\to \mathbb{R}^n$ be a multichannel recording.$~~~~~~~~~~~~~~~~~~$$~~~~~~~~~~~~~~~~$
• Given a window size $\omega$, we compute for every $t\in [\omega,T]$ the embedding $\mathrm{SWE}_{\varphi, \omega}(t) :=\varphi([t-\omega, t])$ in $\mathbb{R}^n$.

• Let $\varphi = (\varphi_1, \varphi_2, \dots \varphi_n):[0,T]\to \mathbb{R}^n$ be a multichannel recording.$~~~~~~~~~~~~~~~~~~$$~~~~~~~~~~~~~~~~$
• Given a window size $\omega$, we compute for every $t\in [\omega,T]$ the embedding $\mathrm{SWE}_{\varphi, \omega}(t) :=\varphi([t-\omega, t])$ in $\mathbb{R}^n$.
• We compute the persistent homology $\mathrm{dgm}(\mathrm{SWE}_{\varphi, \omega}(t))$ of the time evolving sliding window embedding.

• Let $\varphi = (\varphi_1, \varphi_2, \dots \varphi_n):[0,T]\to \mathbb{R}^n$ be a multichannel recording.$~~~~~~~~~~~~~~~~~~$$~~~~~~~~~~~~~~~~$
• Given a window size $\omega$, we compute for every $t\in [\omega,T]$ the embedding $\mathrm{SWE}_{\varphi, \omega}(t) :=\varphi([t-\omega, t])$ in $\mathbb{R}^n$.
• We compute the persistent homology $\mathrm{dgm}(\mathrm{SWE}_{\varphi, \omega}(t))$ of the time evolving sliding window embedding.

Given $\varphi = (\varphi_1, \varphi_2, \dots \varphi_n):[0,T]\to \mathbb{R}^n$ a multichannel recording and $\omega$ a window size.

\[\mathbb{R} ~\longrightarrow ~\mathrm{Top} ~\longrightarrow ~\mathrm{Dgm}~~~~~~~~~~~~~~~~~~~~~~~~~~~~\] \[t\mapsto \mathrm{SWE}_{\varphi, \omega} (t)\mapsto \mathrm{dgm}\big(\mathrm{SWE}_{\varphi, \omega}(t) \big)=:\mathcal{D}_t\]

We compute the following topological summaries:

• Approximate Derivative: Given a time step $\varepsilon>0$\[\dfrac{d_{W}(\mathcal{D}_t, \mathcal{D}_{t-\varepsilon})}{\varepsilon}\]
• Total persistence: \[\sum_{(x,y) \in \mathcal{D_t}} \mathrm{pers}(x,y).\]

• Scalp EEG with 23 sensors at frequency 256 Hz.
• Source: CHB-MIT database from Children’s Hospital Boston, USA.

• Intracraneal EEG with 9 sensors at frequency 200 Hz.
• Source: Clinical presurgical analysis, Toronto Western Hospital, Canada.

• MEG with 144 sensors at frequency 625 Hz
• Source: Toronto Western Hospital, Canada.

• Window size $\omega$ equivalent to 1 or 2 seconds.


• Time step $\varepsilon$ equivalent to 0.5 seconds.


• Execution time (MacBook Pro 8-core CPU 32-core GPU 16 GB unified memory):
  • Persistence diagrams (Ripser) $\sim 0.25$ seconds
  • 1-Wasserstein distance (Ripser) $\sim 0.1$ seconds
  • Total persistence $\sim 0.5\times 10^{-5}$ seconds.


• The computations can be performed in real time.

• Focal seizures, involve part of the cerebral hemisphere.


• Generalized seizures, involve the entire brain, usually presenting loss of consciousness.

Goal: Topologically understand how local vs global dynamics changes over the time.

The time delay embedding of $\varphi_i$ with parameters $d$ and $\tau$ is the vector-valued function \begin{align} X_{d,\tau}\varphi_i\colon &~[0, T-d\tau]\rightarrow ~~\mathbb{R}^{d+1}\\ &~~~~~~~~~~~~~~~t ~~\mapsto ~(\varphi_i(t), \varphi_i(t + \tau), \varphi_i(t + 2\tau), \dots, \varphi_i(t + d\tau))\\ \end{align}

Given a single time series data $\varphi_i(t)$ observed from a (potentially unknown) dynamical system $(X, \phi)$, Takens’ theorem implies that (generically) the time delay embedding $X_{d,\tau}\varphi_i([0,T])$ provides a topological copy of $\{\phi(t, x_0) ∶ t \in [0,T]\} \subseteq X$ for a start point $x_0\in X$.

• Automated detection.
  $~~~~$It is possible to topologically and computationally detect epileptic seizures by means of time-evolving persistence diagrams.

• Automated detection.
  $~~~~$It is possible to topologically and computationally detect epileptic seizures by means of time-evolving persistence diagrams.


• Universality.
  $~~~~$The topological biomarkers are sensitive to epileptic activity under a variety of settings: different methods of recording (MEG, EEG, iEEG), number of sensors and types of epilepsy.

• Automated detection.
  $~~~~$It is possible to topologically and computationally detect epileptic seizures by means of time-evolving persistence diagrams.


• Universality.
  $~~~~$The topological biomarkers are sensitive to epileptic activity under a variety of settings: different methods of recording (MEG, EEG, iEEG), number of sensors and types of epilepsy.


• Robustness.
  $~~~~$The topological summaries are (more) robust to noisy recordings (stability).

• Automated detection.
  $~~~~$It is possible to topologically and computationally detect epileptic seizures by means of time-evolving persistence diagrams.


• Universality.
  $~~~~$The topological biomarkers are sensitive to epileptic activity under a variety of settings: different methods of recording (MEG, EEG, iEEG), number of sensors and types of epilepsy.


• Robustness.
  $~~~~$The topological summaries are (more) robust to noisy recordings (stability).


• Real-time analysis.
  $~~~~$ This is a unsupervised method (no preprocessing or training step), which updates every $\varepsilon$ seconds (for reasonable choices of $\omega$ and $\varepsilon$) and can be computed in real time.

Next step. Explain the connection between models of epileptic seizures (in terms of oscillatory network dynamics) and the topological biomarkers.