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Harding S, Leishman Q, Lunceford W, Passey DJ, Pool T, Webb B. Global forecasts in reservoir computers. Chaos 2024; 34:023136. [PMID: 38407397 DOI: 10.1063/5.0181694] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/17/2023] [Accepted: 01/24/2024] [Indexed: 02/27/2024]
Abstract
A reservoir computer is a machine learning model that can be used to predict the future state(s) of time-dependent processes, e.g., dynamical systems. In practice, data in the form of an input-signal are fed into the reservoir. The trained reservoir is then used to predict the future state of this signal. We develop a new method for not only predicting the future dynamics of the input-signal but also the future dynamics starting at an arbitrary initial condition of a system. The systems we consider are the Lorenz, Rossler, and Thomas systems restricted to their attractors. This method, which creates a global forecast, still uses only a single input-signal to train the reservoir but breaks the signal into many smaller windowed signals. We examine how well this windowed method is able to forecast the dynamics of a system starting at an arbitrary point on a system's attractor and compare this to the standard method without windows. We find that the standard method has almost no ability to forecast anything but the original input-signal while the windowed method can capture the dynamics starting at most points on an attractor with significant accuracy.
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Affiliation(s)
- S Harding
- Mathematics Department, Brigham Young University, Provo, Utah 84602, USA
| | - Q Leishman
- Mathematics Department, Brigham Young University, Provo, Utah 84602, USA
| | - W Lunceford
- Mathematics Department, Brigham Young University, Provo, Utah 84602, USA
| | - D J Passey
- Mathematics Department, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, USA
| | - T Pool
- The Robotics Institute, Carnegie Mellon University, Pittsburg, Pennsylvania 15289, USA
| | - B Webb
- Mathematics Department, Brigham Young University, Provo, Utah 84602, USA
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Huang WA, Stitt IM, Negahbani E, Passey DJ, Ahn S, Davey M, Dannhauer M, Doan TT, Hoover AC, Peterchev AV, Radtke-Schuller S, Fröhlich F. Transcranial alternating current stimulation entrains alpha oscillations by preferential phase synchronization of fast-spiking cortical neurons to stimulation waveform. Nat Commun 2021; 12:3151. [PMID: 34035240 PMCID: PMC8149416 DOI: 10.1038/s41467-021-23021-2] [Citation(s) in RCA: 44] [Impact Index Per Article: 14.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/09/2019] [Accepted: 03/24/2021] [Indexed: 12/12/2022] Open
Abstract
Computational modeling and human studies suggest that transcranial alternating current stimulation (tACS) modulates alpha oscillations by entrainment. Yet, a direct examination of how tACS interacts with neuronal spiking activity that gives rise to the alpha oscillation in the thalamo-cortical system has been lacking. Here, we demonstrate how tACS entrains endogenous alpha oscillations in head-fixed awake ferrets. We first show that endogenous alpha oscillations in the posterior parietal cortex drive the primary visual cortex and the higher-order visual thalamus. Spike-field coherence is largest for the alpha frequency band, and presumed fast-spiking inhibitory interneurons exhibit strongest coupling to this oscillation. We then apply alpha-tACS that results in a field strength comparable to what is commonly used in humans (<0.5 mV/mm). Both in these ferret experiments and in a computational model of the thalamo-cortical system, tACS entrains alpha oscillations by following the theoretically predicted Arnold tongue. Intriguingly, the fast-spiking inhibitory interneurons exhibit a stronger entrainment response to tACS in both the ferret experiments and the computational model, likely due to their stronger endogenous coupling to the alpha oscillation. Our findings demonstrate the in vivo mechanism of action for the modulation of the alpha oscillation by tACS.
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Affiliation(s)
- Wei A Huang
- Department of Psychiatry, University of North Carolina, Chapel Hill, NC, USA
- Carolina Center for Neurostimulation, University of North Carolina, Chapel Hill, NC, USA
- Neuroscience Center, University of North Carolina, Chapel Hill, NC, USA
| | - Iain M Stitt
- Department of Psychiatry, University of North Carolina, Chapel Hill, NC, USA
- Carolina Center for Neurostimulation, University of North Carolina, Chapel Hill, NC, USA
| | - Ehsan Negahbani
- Department of Psychiatry, University of North Carolina, Chapel Hill, NC, USA
- Carolina Center for Neurostimulation, University of North Carolina, Chapel Hill, NC, USA
| | - D J Passey
- Carolina Center for Neurostimulation, University of North Carolina, Chapel Hill, NC, USA
- Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA
| | - Sangtae Ahn
- Department of Psychiatry, University of North Carolina, Chapel Hill, NC, USA
- Carolina Center for Neurostimulation, University of North Carolina, Chapel Hill, NC, USA
- School of Electronic and Electrical Engineering, Kyungpook National University, Daegu, South Korea
| | - Marshall Davey
- Department of Psychiatry, University of North Carolina, Chapel Hill, NC, USA
- Neuroscience Center, University of North Carolina, Chapel Hill, NC, USA
| | - Moritz Dannhauer
- Department of Psychiatry and Behavioral Science, Duke University, Durham, NC, USA
| | - Thien T Doan
- Department of Psychiatry and Behavioral Science, Duke University, Durham, NC, USA
- Department of Biomedical Engineering, Duke University, Durham, NC, USA
| | - Anna C Hoover
- Department of Biomedical Engineering, Duke University, Durham, NC, USA
| | - Angel V Peterchev
- Department of Psychiatry and Behavioral Science, Duke University, Durham, NC, USA
- Department of Biomedical Engineering, Duke University, Durham, NC, USA
- Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA
- Department of Neurosurgery, Duke University, Durham, NC, USA
| | - Susanne Radtke-Schuller
- Department of Psychiatry, University of North Carolina, Chapel Hill, NC, USA
- Carolina Center for Neurostimulation, University of North Carolina, Chapel Hill, NC, USA
| | - Flavio Fröhlich
- Department of Psychiatry, University of North Carolina, Chapel Hill, NC, USA.
- Carolina Center for Neurostimulation, University of North Carolina, Chapel Hill, NC, USA.
- Neuroscience Center, University of North Carolina, Chapel Hill, NC, USA.
- Department of Cell Biology and Physiology, University of North Carolina, Chapel Hill, NC, USA.
- Department of Biomedical Engineering, University of North Carolina, Chapel Hill, NC, USA.
- Department of Neurology, University of North Carolina, Chapel Hill, NC, USA.
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