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uncategorized:ai_safety_arguments_affected_by_chaos:chaos_in_humans [2023/03/31 23:27]
jeffreyheninger created |
uncategorized:ai_safety_arguments_affected_by_chaos:chaos_in_humans [2023/04/07 20:48] (current)
jeffreyheninger |
| ==== Background ==== | ==== Background ==== |
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| If humans are chaotic, then our long-time behavior is inherently unpredictable. Not even a superintelligence would be able to reliably predict the behavior of a particular human. The strong version of this claim is clearly false: there are some things that we can predict about human behavior, even long in advance. We expect these to be associated with non-chaotic dynamics in the brain. The weak claim is more interesting. Does there exist some dynamics in the human brain which is chaotic? Are there some aspects of human behavior that are inherently unpredictable? | If humans are chaotic, then our long-time behavior is inherently unpredictable. Not even a superintelligence would be able to reliably predict the behavior of a particular human. |
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| The answer is less clear than we had hoped at the beginning of the investigation, but it seems as though chaos is at least possible on all or almost all of the scales relevant to human behavior, from atoms to societies. This page largely follows a review article from 2003 by Korn & Faure which addresses this question.((Korn & Faure. //Is there chaos in the brain? II. Experimental evidence and related models.// Comptes Rendus Biologies **326.9**. (2003) p. 787-840. [[https://www.sciencedirect.com/science/article/pii/S1631069103002002]].)) | The strong version of this claim is clearly false: there are some things that we can predict about human behavior, even long in advance. We expect these to be associated with non-chaotic dynamics in the brain. The weak claim is more interesting. Does there exist some dynamics in the human brain which is chaotic? Are there some aspects of human behavior that are inherently unpredictable? |
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| | The answer is less clear than we had hoped at the beginning of the investigation, but it seems as though chaos is at least possible on all or almost all of the scales relevant to human behavior. |
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| | There are various scales on which humans might be chaotic. At the largest scales, are human societies chaotic? At the human scale, how chaotic or predictable is an individual’s behavior? At the organ scale, how chaotic is the human brain? Other organs might also exhibit chaos, but it seems less likely that chaos in the endocrine system, for example, would be as important for understanding human behavior as chaos in the brain. At the cellular scale, do individual neurons behave chaotically? At the atomic and molecular scales, thermal noise and quantum effects dominate. This is not classical chaos, but it does provide the microscopic uncertainty that might be amplified by the chaos at larger scales. |
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| | It seems as though, at most scales, there can be either chaotic or non-chaotic behavior, and there can be transitions between the two.((This is sometimes expressed as the Critical Brain Hypothesis: the brain, and many other biological systems, operate close to the transition between chaotic and non-chaotic motion. \\ Bak. //How Nature Works: The Science of Self-Organised Criticality.// (Copernicus Press, New York, 1996).)) This makes it difficult to determine if there is a continual chain of chaos from atomic to macroscopic scales. Even though this is hard to show, it seems likely that this would be true for at least some important sub-system of the brain in some circumstances. Chaos is a thing that a brain can do, even if not everything the brain does is chaotic. |
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| Even if humans are not predictable, we might still be controllable. In order for you to control a chaotic system, it has to be possible to input a signal of the same sort as any of the sources of uncertainty that the chaos amplifies, and at a speed faster than the Lyapunov time. Whenever the trajectory starts to diverge from the path through the chaos you want it to follow, you have to input some signal to correct the path, before the divergence has gotten too large. | Even if humans are not predictable, we might still be controllable. In order for you to control a chaotic system, it has to be possible to input a signal of the same sort as any of the sources of uncertainty that the chaos amplifies, and at a speed faster than the Lyapunov time. Whenever the trajectory starts to diverge from the path through the chaos you want it to follow, you have to input some signal to correct the path, before the divergence has gotten too large. |
| If thermal or quantum uncertainty inside the brain were amplified by chaos at all scales, then it would become macroscopically important. This would imply that humans are uncontrollable as well as chaotic, because the signal you would need to input to correct for this uncertainty would be something atomic-scale somewhere inside the brain. If there were instead some intermediate mesoscale which were not chaotic, then it would be possible to average over the chaos at smaller scales (such as over atoms or proteins) to get a predictable result (e.g. at the level of neural firing). This would shield any macroscopic chaos from the internal microscopic sources of uncertainty. The important sources of uncertainty would then be from the external environment, which is potentially controllable by an external actor - unless the external environment is itself unavoidably chaotic and uncontrollable. | If thermal or quantum uncertainty inside the brain were amplified by chaos at all scales, then it would become macroscopically important. This would imply that humans are uncontrollable as well as chaotic, because the signal you would need to input to correct for this uncertainty would be something atomic-scale somewhere inside the brain. If there were instead some intermediate mesoscale which were not chaotic, then it would be possible to average over the chaos at smaller scales (such as over atoms or proteins) to get a predictable result (e.g. at the level of neural firing). This would shield any macroscopic chaos from the internal microscopic sources of uncertainty. The important sources of uncertainty would then be from the external environment, which is potentially controllable by an external actor - unless the external environment is itself unavoidably chaotic and uncontrollable. |
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| There are various scales on which humans might be chaotic. At the largest scales, are human societies chaotic? At the human scale, how chaotic or predictable is an individual’s behavior? At the organ scale, how chaotic is the human brain? Other organs might also exhibit chaos, but it seems less likely that chaos in the endocrine system, for example, would be as important for understanding human behavior as chaos in the brain. At the cellular scale, do individual neurons behave chaotically? At the atomic and molecular scales, thermal noise and quantum effects dominate. This is not classical chaos, but it does provide the microscopic uncertainty that might be amplified by the chaos at larger scales. | This page largely follows a review article from 2003 by Korn & Faure, supplemented with more recent evidence, which discusses experimental evidence for chaos in the brain at various scales.((Korn & Faure. //Is there chaos in the brain? II. Experimental evidence and related models.// Comptes Rendus Biologies **326.9**. (2003) p. 787-840. [[https://www.sciencedirect.com/science/article/pii/S1631069103002002]].)) |
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| It seems as though, at most scales, there can be either chaotic or non-chaotic behavior, and can transition between the two.((This is sometimes expressed as the Critical Brain Hypothesis: the brain, and many other biological systems, operate close to the transition between chaotic and non-chaotic motion. \\ Bak. //How Nature Works: The Science of Self-Organised Criticality.// (Copernicus Press, New York, 1996).)) This makes it difficult to determine if there is a continual chain of chaos from atomic to macroscopic scales. Even though this is hard to show, it seems likely that this would be true for at least some important sub-system of the brain in some circumstances. Chaos is a thing that a brain can do, even if not everything the brain does is chaotic. | |
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| ==== Atoms ==== | ==== Atoms ==== |
| ==== Macromolecules ==== | ==== Macromolecules ==== |
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| This is the scale we know relatively little about, so we welcome comments from anyone with more expertise. | This is a scale we know relatively little about, so we welcome comments from anyone with more expertise. |
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| Macromolecules, like proteins or lipids, are large enough that they are not directly affected by quantum and thermal effects. The question is whether there exist some important macromolecules in the brain which amplify quantum or thermal uncertainty to larger scales. There seem to be a few examples from elsewhere in biology where this is the case.((Lambert et al. //Quantum Biology.// Nature Physics **9**. (2013) [[https://quantum.ch.ntu.edu.tw/ycclab/wp-content/uploads/2015/01/Nat-Phys-2013-Lambert.pdf]].)) | Macromolecules, like proteins or lipids, are large enough that they are not directly affected by quantum and thermal effects. The question is whether there exist some important macromolecules in the brain which amplify quantum or thermal uncertainty to larger scales. There seem to be a few examples from elsewhere in biology where this is the case.((Lambert et al. //Quantum Biology.// Nature Physics **9**. (2013) [[https://quantum.ch.ntu.edu.tw/ycclab/wp-content/uploads/2015/01/Nat-Phys-2013-Lambert.pdf]].)) |
| In order for photosynthesis to convert sunlight into chemical energy, light has to first be absorbed by a structure which acts as an antenna, then the excitation has to be transferred to a reaction center, where it can be used for charge separation. This transfer of the excitation occurs with almost perfect efficiency; it is more efficient than is possible classically. Instead, quantum coherence helps the electron find the quickest path to the reaction center, despite the warm and wet biological setting.((Philip Ball. //Is photosynthesis quantum-ish?// Physics World. (2018) [[https://physicsworld.com/a/is-photosynthesis-quantum-ish/]].)) | In order for photosynthesis to convert sunlight into chemical energy, light has to first be absorbed by a structure which acts as an antenna, then the excitation has to be transferred to a reaction center, where it can be used for charge separation. This transfer of the excitation occurs with almost perfect efficiency; it is more efficient than is possible classically. Instead, quantum coherence helps the electron find the quickest path to the reaction center, despite the warm and wet biological setting.((Philip Ball. //Is photosynthesis quantum-ish?// Physics World. (2018) [[https://physicsworld.com/a/is-photosynthesis-quantum-ish/]].)) |
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| Some bird species can sense the earth’s magnetic field and use it to navigate.((This is discussed some more in the blog post [[https://aiimpacts.org/whole-bird-emulation-requires-quantum-mechanics/|Whole Bird Emulation requires Quantum Mechanics]].)) The mechanism seems to involve quantum spin states of pairs of electrons.((Holland. //True navigation in birds: from quantum physics to global migration.// Journal of Zoology **293**. (2014) [[https://zslpublications.onlinelibrary.wiley.com/doi/pdfdirect/10.1111/jzo.12107]].)) In a bird’s retina, there are some pigments (cryptochromes) which absorb light and separate electrons into a radical pair. The spins of the electrons are initially pointing in opposite directions (in a singlet state), but an external magnetic field can flip them so they point in the same direction (in a triplet state). The decay products of the singlet and triplet are different. The retina can detect how the relative concentration of the decay products changes, allowing the bird to “see” the magnetic field. The energy difference between the two states is associated with a frequency of 1-100 MHz.(( One experiment found the response to be in a much narrower band from 1-3 MHz: \\ Ritz. //Quantum effects in biology: Bird navigation.// Procedia Chemistry **3**. (2011) [[https://www.sciencedirect.com/science/article/pii/S1876619611000738]].)) Exposing a bird (European robin) to a weak magnetic field oscillating at this frequency disorients it, so it is unable to navigate. This sense requires maintaining quantum coherence of the spins for at least 100 microseconds, longer than the best man-made spin states in similarly warm and wet environments.((Gauger et al. //Sustained quantum coherence and entanglement in the avian compass.// Physical Review Letters **106**. (2011) [[https://arxiv.org/pdf/0906.3725.pdf]].)) | Some bird species can sense the earth’s magnetic field and use it to navigate.((This is discussed some more in the blog post [[https://blog.aiimpacts.org/p/whole-bird-emulation-requires-quantum-mechanics|Whole Bird Emulation requires Quantum Mechanics]].)) The mechanism seems to involve quantum spin states of pairs of electrons.((Holland. //True navigation in birds: from quantum physics to global migration.// Journal of Zoology **293**. (2014) [[https://zslpublications.onlinelibrary.wiley.com/doi/pdfdirect/10.1111/jzo.12107]].)) In a bird’s retina, there are some pigments (cryptochromes) which absorb light and separate electrons into a radical pair. The spins of the electrons are initially pointing in opposite directions (in a singlet state), but an external magnetic field can flip them so they point in the same direction (in a triplet state). The decay products of the singlet and triplet are different. The retina can detect how the relative concentration of the decay products changes, allowing the bird to “see” the magnetic field. The energy difference between the two states is associated with a frequency of 1-100 MHz.(( One experiment found the response to be in a much narrower band from 1-3 MHz: \\ Ritz. //Quantum effects in biology: Bird navigation.// Procedia Chemistry **3**. (2011) [[https://www.sciencedirect.com/science/article/pii/S1876619611000738]].)) Exposing a bird (European robin) to a weak magnetic field oscillating at this frequency disorients it, so it is unable to navigate. This sense requires maintaining quantum coherence of the spins for at least 100 microseconds, longer than the best man-made spin states in similarly warm and wet environments.((Gauger et al. //Sustained quantum coherence and entanglement in the avian compass.// Physical Review Letters **106**. (2011) [[https://arxiv.org/pdf/0906.3725.pdf]].)) |
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| A few other biological quantum processes have been proposed, although they do not seem to have gained as broad of acceptance as these two. Electron transfer over tens of angstroms between redox centers in proteins might occur via tunneling,((Gray & Winkler. //Electron tunneling through proteins.// Quarterly Reviews of Biophysics **36.3**. (2003) [[https://web.archive.org/web/20050312141415id_/http://www.userwebs.pomona.edu:80/~ejc14747/Chem%20180/Gray_Winkler_QRB_et_proteins.pdf]].)) including possible interference between different paths.((Curry et al. //Pathways, Pathway Tubes, Pathway Docking, and Propagators in Electron Transfer Proteins.// Journal of Bioenergetics and Biomembranes **27.3**. (1995) [[http://www.cvri.ucsf.edu/~grabe/papers/Curry(1995).pdf]].)) Biological photoreceptors can be extremely sensitive to small amounts of light. Rods in human eyes respond to single photons,((Rieke & Baylor. //Single-photon detection by rod cells of the retina.// Reviews of Modern Physics **70.3**. (1998) [[https://www.cns.nyu.edu/csh/csh04/Articles/Rieke1998.pdf]].)) although the retina does not seem to send a single to the brain until close to 10 photons have been detected. The sense of smell might partially identify molecules by their vibrational modes excited by inelastic electron tunneling,((Turin. //A Spectroscopic Mechanism for Primary Olfactory Reception.// Chemical Senses **21.6**. (1996) p. 773-791. [[https://academic.oup.com/chemse/article/21/6/773/488342]].)) although this is disputed. | A few other biological quantum processes have been proposed, although they do not seem to have gained as broad of acceptance as these two. Electron transfer over tens of angstroms between redox centers in proteins might occur via tunneling,((Gray & Winkler. //Electron tunneling through proteins.// Quarterly Reviews of Biophysics **36.3**. (2003) [[https://web.archive.org/web/20050312141415id_/http://www.userwebs.pomona.edu:80/~ejc14747/Chem%20180/Gray_Winkler_QRB_et_proteins.pdf]].)) including possible interference between different paths.((Curry et al. //Pathways, Pathway Tubes, Pathway Docking, and Propagators in Electron Transfer Proteins.// Journal of Bioenergetics and Biomembranes **27.3**. (1995) [[http://www.cvri.ucsf.edu/~grabe/papers/Curry(1995).pdf]].)) Biological photoreceptors can be extremely sensitive to small amounts of light. Rods in human eyes respond to single photons,((Rieke & Baylor. //Single-photon detection by rod cells of the retina.// Reviews of Modern Physics **70.3**. (1998) [[https://www.cns.nyu.edu/csh/csh04/Articles/Rieke1998.pdf]].)) although the retina does not seem to send a signal to the brain until close to 10 photons have been detected. The sense of smell might partially identify molecules by their vibrational modes excited by inelastic electron tunneling,((Turin. //A Spectroscopic Mechanism for Primary Olfactory Reception.// Chemical Senses **21.6**. (1996) p. 773-791. [[https://academic.oup.com/chemse/article/21/6/773/488342]].)) although this is disputed. |
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| Biological settings are not places where you would expect quantum coherence to persist for very long. They have too high temperatures and lots of complex interactions. However, there are a few examples of quantum coherence persisting for much larger time or length scales than we would naively think is possible. Additionally, sensory organs can detect and respond to extremely small signals. | Biological settings are not places where you would expect quantum coherence to persist for very long. They have too high temperatures and lots of complex interactions. However, there are a few examples of quantum coherence persisting for much larger time or length scales than we would naively think is possible. Additionally, sensory organs can detect and respond to extremely small signals. |
| ==== Cells ==== | ==== Cells ==== |
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| Neurons are a type of cell in the brain which process information. Neurons are connected in large networks which act collectively to produce behavior. The way they are connected are through synapses, small junctions in between two neurons where neurotransmitters can be exchanged. | Neurons are a type of cell in the brain which process information. Neurons are connected in large networks which act collectively to produce behavior. The way they are connected is through synapses, small junctions in between two neurons where neurotransmitters can be exchanged. |
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| When neurotransmitters are released at synaptic junctions, they can bind to receptors embedded in the neuron's cell wall. If the receptors are activated by the neurotransmitters, they open and allow an influx of potassium and sodium ions (both positively charged) into the cell, which causes a spike in the neuron's internal voltage. If that voltage passes a threshold, an action potential is fired. An action potential then travels down the axon of a neuron, causing the synapses of that neuron to release neurotransmitters to the next neuron, and the cycle repeats (shown below). | When neurotransmitters are released at synaptic junctions, they can bind to receptors embedded in the neuron's cell wall. If the receptors are activated by the neurotransmitters, they open and allow an influx of potassium and sodium ions (both positively charged) into the cell, which causes a spike in the neuron's internal voltage. If that voltage passes a threshold, an action potential is fired. An action potential then travels down the axon of a neuron, causing the synapses of that neuron to release neurotransmitters to the next neuron, and the cycle repeats (shown below). |
| While the Hodgkin-Huxley model was originally for a single patch of the cell membrane, introducing interaction terms between neighboring patches allows it to model the propagation of a signal along the axon. Hindmarsh and Rose reduced this model to a three variable differential equation for the neuron as a whole.((Hindmarsh & Rose. //A model of neuronal bursting using three coupled first order differential equations.// Proceedings of the Royal Society of London. Series B, Biological Sciences **221.1222**. (1984) p. 87-102. [[https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=fe301505cb75744bfa1eb6f26737ffb17a09c493]].)) It also shows bifurcations, bistable behavior, and chaos. Often, the model will show the neurons firing in bursts, with long quiescent periods in between. | While the Hodgkin-Huxley model was originally for a single patch of the cell membrane, introducing interaction terms between neighboring patches allows it to model the propagation of a signal along the axon. Hindmarsh and Rose reduced this model to a three variable differential equation for the neuron as a whole.((Hindmarsh & Rose. //A model of neuronal bursting using three coupled first order differential equations.// Proceedings of the Royal Society of London. Series B, Biological Sciences **221.1222**. (1984) p. 87-102. [[https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=fe301505cb75744bfa1eb6f26737ffb17a09c493]].)) It also shows bifurcations, bistable behavior, and chaos. Often, the model will show the neurons firing in bursts, with long quiescent periods in between. |
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| Several other experiments on single neurons have also exhibited chaos. Individual buccal-oral neurons of the sea slug Pleurobranchae californica showed evidence of chaos, although the time series was short and non-stationary,((Mpitsos. //Evidence for chaos in spike trains of neurons that generate rythmic motor patterns.// Brain Research Bulletin **21.3**. (1988) p. 529-538. [[https://www.sciencedirect.com/science/article/abs/pii/0361923088901694]].)) while the K15 neuron of another mollusc Aplysia californica showed bistablility and irregular & bursting firing patterns.((Frazier et al. //Morphological and functional properties of identified neurons in the abdominal ganglion of Aplysia Californica.// Journal of Neurophysiology **30.6**. (1967) p. 1299-1351. [[https://journals.physiology.org/doi/pdf/10.1152/jn.1967.30.6.1288]].)) The anterior burster neuron of the stomatogastric ganglion of the spiny lobster Pancibirus interruptus operates close to a bifurcation point which allows it to transition between periodic and chaotic motion.((Guckenheimer et al. //Mapping the dynamics of a bursting neuron.// Proceedings of the Royal Society of London. Series B, Biological Science. (1993) [[https://royalsocietypublishing.org/doi/pdf/10.1098/rstb.1993.0121]].)) Individual neurons also seem to be capable of distinguishing between chaotic and stochastic signals: sensory neurons from a rat’s skin showed observably different patterns of spikes in response to a stochastic input, compared to an input derived from the Rössler equations.((Richardson et al. //Encoding chaos in neural spike trains.// Physical Review Letters **80.11**. (1998) [[https://www.researchgate.net/profile/Peter-Grigg/publication/239967347_Encoding_Chaos_in_Neural_Spike_Trains/links/56b58e0608ae3c1b79ab24ea/Encoding-Chaos-in-Neural-Spike-Trains.pdf]].)) | Several other experiments on single neurons have also exhibited chaos. Individual buccal-oral neurons of the sea slug //Pleurobranchae californica// showed evidence of chaos, although the time series was short and non-stationary,((Mpitsos. //Evidence for chaos in spike trains of neurons that generate rythmic motor patterns.// Brain Research Bulletin **21.3**. (1988) p. 529-538. [[https://www.sciencedirect.com/science/article/abs/pii/0361923088901694]].)) while the K15 neuron of another mollusc //Aplysia californica// showed bistablility and irregular & bursting firing patterns.((Frazier et al. //Morphological and functional properties of identified neurons in the abdominal ganglion of Aplysia Californica.// Journal of Neurophysiology **30.6**. (1967) p. 1299-1351. [[https://journals.physiology.org/doi/pdf/10.1152/jn.1967.30.6.1288]].)) The anterior burster neuron of the stomatogastric ganglion of the spiny lobster //Pancibirus interruptus// operates close to a bifurcation point which allows it to transition between periodic and chaotic motion.((Guckenheimer et al. //Mapping the dynamics of a bursting neuron.// Proceedings of the Royal Society of London. Series B, Biological Science. (1993) [[https://royalsocietypublishing.org/doi/pdf/10.1098/rstb.1993.0121]].)) Individual neurons also seem to be capable of distinguishing between chaotic and stochastic signals: sensory neurons from a rat’s skin showed observably different patterns of spikes in response to a stochastic input, compared to an input derived from the Rössler equations.((Richardson et al. //Encoding chaos in neural spike trains.// Physical Review Letters **80.11**. (1998) [[https://www.researchgate.net/profile/Peter-Grigg/publication/239967347_Encoding_Chaos_in_Neural_Spike_Trains/links/56b58e0608ae3c1b79ab24ea/Encoding-Chaos-in-Neural-Spike-Trains.pdf]].)) |
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| It seems pretty common, both in theoretical models and in experiments, for individual neurons to behave chaotically. Chaos does not just occur in pathological conditions or environments far from physiologically realistic: individual neurons can behave chaotically in conditions similar to those in living organisms. This suggests that the normal behavior of neurons sometimes makes use of the chaotic behavior which can occur in individual cells. The observation that individual neurons can distinguish between chaotic and stochastic signals further suggests that chaotic behavior is important, because if it were not, there would be no reason for neurons to have this ability. | It seems pretty common, both in theoretical models and in experiments, for individual neurons to behave chaotically. Chaos does not just occur in pathological conditions or environments far from physiologically realistic: individual neurons can behave chaotically in conditions similar to those in living organisms. This suggests that the normal behavior of neurons sometimes makes use of the chaotic behavior which can occur in individual cells. The observation that individual neurons can distinguish between chaotic and stochastic signals further suggests that chaotic behavior is important, because if it were not, there would be no reason for neurons to have this ability. |
| Two experiments stand out as particularly good examples of how chaos and transitions to and from chaos can play a role in the nervous system. | Two experiments stand out as particularly good examples of how chaos and transitions to and from chaos can play a role in the nervous system. |
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| When shown a flash of light, the neurons in a retina (of tiger salamander larvae or humans) exhibit a particular firing pattern, which stops shortly after the flash.((Crevier & Meister. //Synchronous Period-Doubling in Flicker of Salamander and Man.// Journal of Neurophysiology **79**. (1998) p. 1869-1878. [[https://journals.physiology.org/doi/pdf/10.1152/jn.1998.79.4.1869]].)) If the flash is shown periodically, with a frequency of less than 9 Hz, the retina’s response is periodic: each flash is followed by this particular firing pattern. When the frequency exceeds 9 Hz, there is not enough time for this pattern to complete, so every other response is different. The retina responds periodically, with a period double the period of the flashing light. When the frequency exceeds 12 Hz, the period doubles again and the retina’s response is the same only after every four flashes. When the frequency exceeds 15 Hz, the retina’s response is chaotic. The subjects say that the light appears “flickering.” This is a period doubling cascade, Feigenbaum’s classic route to chaos.((This is described in [[https://docs.google.com/document/d/1HyRd0SyDGIG49vkCKssD2HmPnN8Gbrw1duPXrLaRL9U/edit?usp=sharing|Chaos and Intrinsic Unpredictability]]. See also Wikipedia: [[https://en.wikipedia.org/wiki/Period-doubling_bifurcation|Period-doubling bifurcation]] or Scholarpedia: [[http://www.scholarpedia.org/article/Period_doubling|Period doubling]].)) Feigenbaum’s theory predicts that, with more precise control over the frequency of the flashing light, you could see the retina respond at 8 or 16 times the period of the flashing light, and that the period doubling bifurcations get closer together in a way characterized by Feigenbaum’s constants, before transitioning to chaos. | When shown a flash of light, the neurons in a retina (of tiger salamander larvae or humans) exhibit a particular firing pattern, which stops shortly after the flash.((Crevier & Meister. //Synchronous Period-Doubling in Flicker of Salamander and Man.// Journal of Neurophysiology **79**. (1998) p. 1869-1878. [[https://journals.physiology.org/doi/pdf/10.1152/jn.1998.79.4.1869]].)) If the flash is shown periodically, with a frequency of less than 9 Hz, the retina’s response is periodic: each flash is followed by this particular firing pattern. When the frequency exceeds 9 Hz, there is not enough time for this pattern to complete, so every other response is different. The retina responds periodically, with a period double the period of the flashing light. When the frequency exceeds 12 Hz, the period doubles again and the retina’s response is the same only after every four flashes. When the frequency exceeds 15 Hz, the retina’s response is chaotic. The subjects say that the light appears “flickering.” This is a period doubling cascade, Feigenbaum’s classic route to chaos.((This is described in Section 9 of the accompanying report. \\ Heninger & Johnson. //Chaos and Intrinsic Unpredictability.// AI Impacts. [[http://aiimpacts.org/wp-content/uploads/2023/04/Chaos-and-Intrinsic-Unpredictability.pdf]]. \\ See also Wikipedia: [[https://en.wikipedia.org/wiki/Period-doubling_bifurcation|Period-doubling bifurcation]] or Scholarpedia: [[http://www.scholarpedia.org/article/Period_doubling|Period doubling]].)) Feigenbaum’s theory predicts that, with more precise control over the frequency of the flashing light, you could see the retina respond at 8 or 16 times the period of the flashing light, and that the period doubling bifurcations get closer together in a way characterized by Feigenbaum’s constants, before transitioning to chaos. |
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| The normal firing pattern of neurons in the olfactory bulb of rabbits is chaotic.((Di Prisco & Freeman. //Odor-related bulbar EEG spatial pattern analysis during appetitive conditioning in rabbits.// Behavioral Neuroscience **99.5**. (1985) [[https://escholarship.org/content/qt7s63p7sx/qt7s63p7sx.pdf]]. \\ Freeman & Di Prisco. //Spatial patterns differences with discriminated odors manifest chaotic and limit cycles attractors in olfactory bulb of rabbits.// Brain Theory. (1986) p. 97-119.)) When exposed to a smell the rabbit has previously learned, the firing patterns cease to be chaotic and instead become periodic. The periodic motion seems to follow one of the unstable periodic orbits embedded in the original strange attractor. Each smell the rabbit has previously learned corresponds to a different periodic orbit. It seems as though the olfactory bulb is using a kind of dynamical memory storage, which allows rapid responses to learned stimuli. The smells are remembered as unstable periodic orbits within the strange attractor. | The normal firing pattern of neurons in the olfactory bulb of rabbits is chaotic.((Di Prisco & Freeman. //Odor-related bulbar EEG spatial pattern analysis during appetitive conditioning in rabbits.// Behavioral Neuroscience **99.5**. (1985) [[https://escholarship.org/content/qt7s63p7sx/qt7s63p7sx.pdf]]. \\ Freeman & Di Prisco. //Spatial patterns differences with discriminated odors manifest chaotic and limit cycles attractors in olfactory bulb of rabbits.// Brain Theory. (1986) p. 97-119.)) When exposed to a smell the rabbit has previously learned, the firing patterns cease to be chaotic and instead become periodic. The periodic motion seems to follow one of the unstable periodic orbits embedded in the original strange attractor. Each smell the rabbit has previously learned corresponds to a different periodic orbit. It seems as though the olfactory bulb is using a kind of dynamical memory storage, which allows rapid responses to learned stimuli. The smells are remembered as unstable periodic orbits within the strange attractor. |
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| Similar experiments have been done on anesthetized spider monkeys,((Rapp et al. //Dynamics of spontaneous neural activity in the simian motor cortex: the dimension of chaotic neurons.// Physics Letters A **110.6**. (1985) p. 335-338. [[https://www.sciencedirect.com/science/article/abs/pii/0375960185907868]].)) anesthetized rats,((Celletti & Villa. //Low-dimensional chaotic attractors in the rat brain.// Biological Cybernetics **74**. (1996) p. 387-393. [[https://link.springer.com/article/10.1007/BF00206705]].)) asleep cats,((Röschke & Basar. //Dynamics of Sensory and Cognitive Processing by the Brain.// Ch. 4: The EEG is not a simple noise: strange attractors in intracranial structures. (Springer-Verlag, Berlin, 1988))) and awake chickens.((Neuenschwander et al. //A dynamical analysis of oscillatory responses in the optic tectum.// Cognitive Brain Research **1.3**. (1993) p. 175-181. [[https://www.sciencedirect.com/science/article/abs/pii/092664109390025Z]].)) The normal behavior of these sensory neurons seems to be chaotic, and the chaos is altered dramatically in response to a stimulus. | Similar experiments have been done on anesthetized spider monkeys,((Rapp et al. //Dynamics of spontaneous neural activity in the simian motor cortex: the dimension of chaotic neurons.// Physics Letters A **110.6**. (1985) p. 335-338. [[https://www.sciencedirect.com/science/article/abs/pii/0375960185907868]].)) anesthetized rats,((Celletti & Villa. //Low-dimensional chaotic attractors in the rat brain.// Biological Cybernetics **74**. (1996) p. 387-393. [[https://link.springer.com/article/10.1007/BF00206705]].)) asleep cats,((Röschke & Basar. //Dynamics of Sensory and Cognitive Processing by the Brain.// Ch. 4: The EEG is not a simple noise: strange attractors in intracranial structures. (Springer-Verlag, Berlin, 1988) )) and awake chickens.((Neuenschwander et al. //A dynamical analysis of oscillatory responses in the optic tectum.// Cognitive Brain Research **1.3**. (1993) p. 175-181. [[https://www.sciencedirect.com/science/article/abs/pii/092664109390025Z]].)) The normal behavior of these sensory neurons seems to be chaotic, and the chaos is altered dramatically in response to a stimulus. |
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| ==== Entire Brain ==== | ==== Entire Brain ==== |
| Central neurons which have lots of presynaptic cells are particularly interesting. The effect of having this many presynaptic cells influencing them is a continuous variation of the membrane potential, called “synaptic noise”.((Yarom & Hounsgaard. //Voltage Fluctuations in Neurons: Signal or Noise?// Physiological Reviews **91.3**. (2011) p. 917-929. [[https://journals.physiology.org/doi/pdf/10.1152/physrev.00019.2010]].)) This “noise” is likely not simply noise, but instead is a complicated chaotic signal which contains information about various parts of the brain.((Ferstser. //Is Neural Noise Just a Nuisance?// Science **273.5283**. (1996) p. 1812. [[https://www.science.org/doi/abs/10.1126/science.273.5283.1812]].)) These central neurons seem like they would be particularly good at amplifying smaller scale uncertainties. | Central neurons which have lots of presynaptic cells are particularly interesting. The effect of having this many presynaptic cells influencing them is a continuous variation of the membrane potential, called “synaptic noise”.((Yarom & Hounsgaard. //Voltage Fluctuations in Neurons: Signal or Noise?// Physiological Reviews **91.3**. (2011) p. 917-929. [[https://journals.physiology.org/doi/pdf/10.1152/physrev.00019.2010]].)) This “noise” is likely not simply noise, but instead is a complicated chaotic signal which contains information about various parts of the brain.((Ferstser. //Is Neural Noise Just a Nuisance?// Science **273.5283**. (1996) p. 1812. [[https://www.science.org/doi/abs/10.1126/science.273.5283.1812]].)) These central neurons seem like they would be particularly good at amplifying smaller scale uncertainties. |
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| One measurement scientists use to study whole brain behavior is EEG (electroencephalogram). Many electrodes are attached to the skull and electrical activity is recorded. This method is not fine-grained enough to capture electrical activity of single neurons, instead each electrode records from millions of neurons at once. Often, even at this coarse-grained level, clear dynamics emerge. For instance, typically there are oscillations of various frequencies in the brain, such as during periods of concentration or sleep. | One measurement scientists use to study whole brain behavior is EEG (electroencephalogram). Many electrodes are attached to the skull and electrical activity is recorded. This method is not fine-grained enough to capture electrical activity of single neurons; instead each electrode records from millions of neurons at once. Often, even at this coarse-grained level, clear dynamics emerge. For instance, typically there are oscillations of various frequencies in the brain, such as during periods of concentration or sleep. |
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| There is significant disagreement in the literature about whether data from EEGs of normal brain activity exhibit chaos. Some papers show indicators of chaos,((Babloyantz et al. //Evidence of Chaotic Dynamics of Brain Activity during the Sleep Cycle.// Physics Letters **111A.3**. (1985) p. 152-156. [[https://orfeo.belnet.be/bitstream/handle/internal/5872/Babloyantz%281985a%29.pdf?sequence=1&isAllowed=y]].)) while others show that similar behavior can arise from certain types of noise.((Biswal & Dasgupta. //Stochastic neural network model for spontaneous bursting in hippocampal slices.// Physical Review E **66**. (2002) [[https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=c286ddfd6927e4a909c8dcc973ea0d37f52615f1]].)) If normal EEG readings are chaotic, then the chaos is high dimensional, which means that many variables are needed to characterize the motion. This makes it difficult to reconstruct the state space from a time series. If the normal EEG readings show noise, then it is unclear what the source of the noise is - and one possibility is a hard to characterize chaotic system.((Faure & Korn. //Is there chaos in the brain? I. Concepts of nonlinear dynamics and methods of investigation.// Comptes Rendus de l'Académie des Sciences - Series III - Sciences de la Vie **324.9**. (2001) p. 773-793. [[https://www.sciencedirect.com/science/article/abs/pii/S0764446901013774]]. \\ | There is significant disagreement in the literature about whether data from EEGs of normal brain activity exhibit chaos. Some papers show indicators of chaos,((Babloyantz et al. //Evidence of Chaotic Dynamics of Brain Activity during the Sleep Cycle.// Physics Letters **111A.3**. (1985) p. 152-156. [[https://orfeo.belnet.be/bitstream/handle/internal/5872/Babloyantz%281985a%29.pdf?sequence=1&isAllowed=y]].)) while others show that similar behavior can arise from certain types of noise.((Biswal & Dasgupta. //Stochastic neural network model for spontaneous bursting in hippocampal slices.// Physical Review E **66**. (2002) [[https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=c286ddfd6927e4a909c8dcc973ea0d37f52615f1]].)) If normal EEG readings are chaotic, then the chaos is high dimensional, which means that many variables are needed to characterize the motion. This makes it difficult to reconstruct the state space from a time series. If the normal EEG readings show noise, then it is unclear what the source of the noise is - and one possibility is a hard to characterize chaotic system.((Faure & Korn. //Is there chaos in the brain? I. Concepts of nonlinear dynamics and methods of investigation.// Comptes Rendus de l'Académie des Sciences - Series III - Sciences de la Vie **324.9**. (2001) p. 773-793. [[https://www.sciencedirect.com/science/article/abs/pii/S0764446901013774]]. \\ |
| Pritchard et al. //Dimensional analysis of resting human EEG II: Surrogate-data testing indicates nonlinearity but not low-dimensional chaos.// Psychophisiology **32.5**. (1995) p. 486-491. [[https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1469-8986.1995.tb02100.x]].)) Either the noise or the chaos seem like significant barriers to predicting the behavior of the brain. | Pritchard et al. //Dimensional analysis of resting human EEG II: Surrogate-data testing indicates nonlinearity but not low-dimensional chaos.// Psychophisiology **32.5**. (1995) p. 486-491. [[https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1469-8986.1995.tb02100.x]].)) Either the noise or the chaos seem like significant barriers to predicting the behavior of the brain. |
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| EEG data during an epileptic seizure is much less ambiguous. It is clearly chaotic, with a low enough dimensional strange attractor to be clearly reconstructed using time-delay coordinates.((Babloyantz & Destexhe. //Low-dimensional chaos in an instance of epilepsy.// Proceedings of the National Academy of Sciences, USA **83**. (1986) p. 3513-3517. [[https://www.pnas.org/doi/pdf/10.1073/pnas.83.10.3513]].)) Chaos theory can help predict these seizures: the measured Lyapunov exponent, averaged over the time series, starts to fall about minutes before the onset of the seizure.((Iasemidis & Sackellares. //The evolution with time of the spatial distribution of the largest Lyapunov exponent on the human epileptic cortex.// Measuring chaos in the human brain. (1991) p. 49-82. [[https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=f6ebcf1186cacdb62c26f6e81071261fd05daa9c]].)) During the seizure, the Lyapunov time is 0.3-0.5 seconds/bit.((This is equivalent to saying that the uncertainty doubles every 0.3-0.5 seconds.)) It seems as though epilepsy is a dynamical disease, characterized by a reduction in the complexity of the dynamics of the brain, from high dimensional chaos/noise to low dimensional chaos.((da Silva et al. //Epilepsies as Dynamical Diseases of Brain Systems: Basic Models of the Transition Between Normal and Epileptic Activity.// Epilesia **44**. (2003) p. 72-83. [[https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.0013-9580.2003.12005.x]].)) | EEG data during an epileptic seizure is much less ambiguous. It is clearly chaotic, with a low enough dimensional strange attractor to be clearly reconstructed using time-delay coordinates.((Babloyantz & Destexhe. //Low-dimensional chaos in an instance of epilepsy.// Proceedings of the National Academy of Sciences, USA **83**. (1986) p. 3513-3517. [[https://www.pnas.org/doi/pdf/10.1073/pnas.83.10.3513]].)) Chaos theory can help predict these seizures: the measured Lyapunov exponent, averaged over the time series, starts to fall about minutes before the onset of the seizure.((Iasemidis & Sackellares. //The evolution with time of the spatial distribution of the largest Lyapunov exponent on the human epileptic cortex.// Measuring chaos in the human brain. (1991) p. 49-82. [[https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=f6ebcf1186cacdb62c26f6e81071261fd05daa9c]].)) During the seizure, the Lyapunov time is 0.3-0.5 seconds / bit.((This is equivalent to saying that the uncertainty doubles every 0.3-0.5 seconds.)) It seems as though epilepsy is a dynamical disease, characterized by a reduction in the complexity of the dynamics of the brain, from high dimensional chaos/noise to low dimensional chaos.((da Silva et al. //Epilepsies as Dynamical Diseases of Brain Systems: Basic Models of the Transition Between Normal and Epileptic Activity.// Epilesia **44**. (2003) p. 72-83. [[https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.0013-9580.2003.12005.x]].)) |
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| At the scale of the entire brain, there is again a somewhat ambiguous result. There certainly are circumstances where chaos is relevant, but it is hard to tell the extent to which chaos is normal. This is perhaps not surprising to careful students of human behavior. Humans can be both very chaotic and very predictable, with rapid and often unpredictable transitions between the two. | At the scale of the entire brain, there is again a somewhat ambiguous result. There certainly are circumstances where chaos is relevant, but it is hard to tell the extent to which chaos is normal. This is perhaps not surprising to careful students of human behavior. Humans can be both very chaotic and very predictable, with rapid and often unpredictable transitions between the two. |
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| There is some evidence of chaos in financial markets.((Connelly. //Chaos theory and the financial markets.// Journal of Financial Planning 9.6. (1996) p. 26-30. \\ | There is some evidence of chaos in financial markets.((Connelly. //Chaos theory and the financial markets.// Journal of Financial Planning 9.6. (1996) p. 26-30. \\ |
| Vaga. //Profiting from Chaos: Using Chaos Theory for Market Timing, Stock Selection, and Option Valuation.// (1994) )) There is reason to be a bit skeptical of these results. Markets seem like intrinsically high dimensional systems where it is hard to distinguish between chaos and noise. They are probably better understood as being anti-inductive. There could be patterns in financial data which are discoverable using techniques from chaos theory. If they were found, someone would figure out a way to make money off of them, and the pattern would go away. | Vaga. //Profiting from Chaos: Using Chaos Theory for Market Timing, Stock Selection, and Option Valuation.// (1994) )) There is reason to be a bit skeptical of these results. Markets seem like intrinsically high dimensional systems where it is hard to distinguish between chaos and noise. They are probably better understood as being anti-inductive. There could be patterns in financial data which are discoverable using techniques from chaos theory. If they were found, someone would likely figure out a way to make money off of them, and the pattern would go away. |
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| Another way to phrase the question of sensitive dependence on initial conditions((Recurrence in human societies seems like a harder condition to satisfy.)) for human societies is: Can a single person have a lasting impact on history? This is a matter of debate among historians, which we will not dig into here. | Another way to phrase the question of sensitive dependence on initial conditions((Recurrence in human societies seems like a harder condition to satisfy.)) for human societies is: Can a single person have a lasting impact on history? This is a matter of debate among historians, which we will not dig into here. |
| When the dynamics is chaotic, it is often non-stationary and multistable. It is not just the details of the future that vary - the qualitative features and statistical properties of the dynamics vary too. Different initial conditions can lead to dramatically different chaotic states - or to states which are not chaotic. | When the dynamics is chaotic, it is often non-stationary and multistable. It is not just the details of the future that vary - the qualitative features and statistical properties of the dynamics vary too. Different initial conditions can lead to dramatically different chaotic states - or to states which are not chaotic. |
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| While this is a complicated result, it is perhaps not surprising. There are some aspects of human behavior which we find easy to predict. But people also sometimes behave in surprising ways. This surprise is not just because we are ignorant of the state of our brains. The dynamics of the brain can be inherently unpredictable, at time scales of less than 1 second,((The largest estimate of the Lyapunov time for something in the brain is similar to the human reaction time. This makes sense because both are measures of how long it takes for the entire brain to respond to a signal or an uncertainty that initially only involves a small part of the nervous system.)) even with as good of knowledge of the state of the brain as possible. Attempting to completely control a brain would not only involve continual inputs at a frequency of more than 1/second, it would also require precise microscopic inputs at at least some locations inside of the brain. Completely predicting or controlling the dynamics of a human brain is not possible. | While this is a complicated result, it is perhaps not surprising. There are some aspects of human behavior which we find easy to predict. But people also sometimes behave in surprising ways. This surprise is not just because we are ignorant of the state of our brains. The dynamics of the brain can be inherently unpredictable, at time scales of less than 1 second,((The largest estimate of the Lyapunov time for something in the brain is similar to the human reaction time. This makes sense because both are measures of how long it takes for the entire brain to respond to a signal or an uncertainty that initially only involves a small part of the nervous system.)) even with as good of knowledge of the state of the brain as possible. Attempting to completely control a brain would not only involve continual inputs at a frequency of more than 1 / second, it would also require precise microscopic inputs at at least some locations inside of the brain. Completely predicting or controlling the dynamics of a human brain is not possible. |
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| Why would chaos exist in the brain? If there were no advantage to having it, we would expect to see less of it. | Why would chaos exist in the brain? If there were no advantage to having it, we would expect to see less of it. |
