Dynamical Systems Modeling in Neuroscience
- 1. Introduction to Computational Neuroscience: Dynamical Systems Modeling Yohan J. John Research Scientist Neural Systems Lab Boston University
- 2. 2 The Plan Some “sociology” of neuroscience What is a computational model? Intro to differential equations A simple example of a dynamical system The “landscape” analogy The Hodgkin-Huxley model Other neuroscience applications of dynamical systems Hands on with some simulations!
- 3. 3 The “Pentagon” of Mind and Brain Sciences
- 4. 4 A spectrum of methodological approaches
- 5. 5 A spectrum of methodological approaches
- 6. 6 A spectrum of methodological approaches
- 7. 7 What is computational modeling? Mathematical/computational methods are used in two broad approaches: Statistical modeling Capturing patterns in data Uses probabilistic reasoning • Regression (curve fitting) • Hypothesis testing (t-tests, ANOVA) Dynamical modeling Capturing time-varying processes Uses deterministic reasoning (in simple cases) • Differential equations • “Phase plane” analysis
- 8. 8 What is computational modeling for? Predicting future (unmeasured) data Understanding causality Helping formulate hypotheses for future experiments “It’s Difficult to Make Predictions, Especially About the Future” - Niels Bohr? Yogi Berra? Mark Twain? Nostradamus? Anonymous?
- 9. 9 What is a model? Mercator projection Lambert equal area projection
- 10. 10 What is a model? “On Exactitude in Science” (1946) by Jorge Luis Borges
- 11. 11 Internal (cognitive) models One of the goals of computational neuroscience and computational cognitive science is to understand how humans and other animals create these internal models.
- 12. 12 What is a model? "All models are wrong, but some are useful." -- attributed to statistician George Box “Useful” can mean different things to different researchers. Generally, a model is used to • simplify the data, capturing the biggest trends and separating out “noise” or random fluctuations • generate predictions • explain how the data arose The geocentric model was useful for predicting data such as eclipses. The heliocentric model was initially no better at prediction than the geocentric model, but was simpler, provided deeper insight, and stimulated further theorizing.
- 13. 13 Dynamical Systems They capture how systems change with respect to time, space, and other variables. Neuroscience examples: • Membrane potential (voltage) • Concentrations of ions (e.g., sodium, potassium, calcium) • Firing rate (spikes per second) • Field potentials (EEG, LFP etc) • BOLD signals (fMRI)
- 14. 14 Why use dynamical systems? Differential equations can be more powerful than statistical fits of specific data sets. Dynamical systems can help us understand the similarity between processes that initially seem quite different. We will see that dynamical systems can also help us use quantitative techniques to understand qualitative properties of systems. Isaac Newton showed that a cannonball fired fast enough from a tall mountain could “fall” all the way around Earth without touching its surface.
- 15. 15 The derivative: capturing rate of change A derivative (a.k.a. differential) represents the rate of change of a quantity with respect to time, space, or some other variable. example: speed is the derivative of distance with respect to time
- 16. 16 The derivative captures the slope Slope:
- 17. 17 Differential equations The simplest possible differential equation (other than zero change!) example: a car moving at constant speed
- 18. 18 Linear growth Exact solution: A constant growth rate
- 19. 19 Exponential growth Exact solution: A growing growth rate: (This is the continuous version of geometric growth.)
- 20. 20 Predator-prey modeling https://globalchange.umich.edu/globalchange1/current/labs/Lab7_PredatorPrey/Pred_Prey.htm A simple system… but no simple (“closed form”) solution! Coupled differential equations:
- 21. 21 The Lotka-Volterra Predator-Prey Model A “toy” model with simplifying assumptions, such as: 1. Prey find ample food at all times. 2. The food supply of predators depends only on number of prey. 3. The rate of population growth depends on its size. 4. The environment does not change. 5. Predators have limitless appetite.
- 23. 23 Perturbations to the system What might happen if there is a sudden drop in the number of foxes?
- 25. 25 Foxes versus rabbits We can plot the population of foxes versus the population of rabbits. The closed loop indicates the oscillatory dynamics. We can also see how the perturbation made the cycle “bigger”.
- 26. 26 State space Any collection of variables or measurements can be understood as a space. A specific set of values for each variable – a possible state of the system – counts as a point in this space. The number of variables is the number of dimensions of the system. For 1D, 2D and 3D systems, we can visualize the state space.
- 27. 27 The “landscape” of a 1D system Izhikevich, 2007: Dynamical Systems in Neuroscience The ball represents the state of the system The initial position of ball could be anywhere The ball moves downwards according to the slope (i.e., derivative) The ball stops at “fixed points”, where slope is zero Fixed points can be stable or unstable stable stable unstable
- 28. 28 The “landscape” of a 2D system With two variables (x and y), we can visualize the dynamics as a vector field An arrow represents the rate of change at the location (x,y) Arrows point to where a ball would move if placed at that point Each diagram of this type is called a “phase plane” Key concepts: attractors, repellers, saddle points Izhikevich, 2007: Dynamical Systems in Neuroscience
- 29. 29 https://www.rei.com/learn/expert-advice/topo-maps-how-to-use.html Unstable points (“repellers”) are like peaks Stable points (“attractors”) are like valleys Saddle nodes are like ridges between peaks The “landscape” of a 2D system
- 30. 30 The “seascape” of a 2D system? A third kind of structure is called a limit cycle – it is an oscillatory pattern Izhikevich, 2007: Dynamical Systems in Neuroscience
- 31. 31 Foxes versus rabbits on the phase plane The lengths of the arrows tell us how large the derivatives are, and therefore how fast the system will change when it is at that point.
- 32. 32 The dynamical systems approach 1. Find differential equations that capture how the variables change over time. 2. Study the special features of the system • Attractors • Repellers • Saddle Points • Limit Cycles • Strange attractors (a chaotic system!) 3. Use the insights to understand the real system and predict data that haven’t been collected yet. (This is the hardest step!) Studying fixed points and other structures of the state space allows us to understand qualitative features of dynamical systems.
- 33. 33 The neuron: key signaling unit of the brain
- 34. 34 The action potential or “spike” The action potential travels along the axon It is not a current but it involves currents (which mostly flow across the membrane) The axon is not like a metal wire https://biology.stackexchange.com/questions/34075/how-to-conceptualize-the-action-potential
- 35. 35 The biophysics of resting All biological cells allow ions to cross their membranes. This flow can be passive or active. Ion channels are selective for specific ions. At equilibrium, the inside of the cell is at a lower potential than the outside. https://courses.washington.edu/conj/membpot/membranepot.htm
- 36. 36 The biophysics of spiking Changing the membrane potential with input – current injection or neurotransmitter binding – can cause a rapid cascade of events that together create a spike or action potential. The key ions involved in this cascade are sodium and potassium. Animated videos help understand these events.
- 37. 37 The Hodgkin-Huxley model Alan Hodgkin and Andrew Huxley derived a highly accurate model of this cascade of events using differential equations. Schwiening, C. J. (2012). A brief historical perspective: Hodgkin and Huxley. The Journal of Physiology, 590(11), 2571-2575. They recorded from the squid giant axon, and fit their model to its dynamics.
- 38. 38 The Hodgkin-Huxley model of action potentials Can you identify which plot is data and which is the model? Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117(4), 500-544.
- 39. 39 Dynamics of ion flow and membrane potential The Hodgkin-Huxley model describes the dynamics of 4 quantities: V: membrane potential (voltage) n: K channel activation m: Na channel activation h: Na channel inactivation m, n and h are “gating” variables that describe the dynamics of the ion channels
- 40. 40 Simulating a spike using the HH model n: K channel activation m: Na channel activation h: Na channel inactivation
- 42. 42 The Hodgkin-Huxley legacy Recall: Hodgkin and Huxley did not know about the structure of ion channels. “The HH model has been amazingly successful in both describing and predicting a large number of neuronal properties. Extensions of this model, incorporating a variety of voltage-dependent channel types beyond the original HH pair, have been very widely used in research throughout the world. However, as HH were themselves well aware, the success of the model does not in itself constitute convincing evidence that the "pictorial" interpretation of the HH equations is a true reflection of the real molecular events. It is therefore very gratifying, although perhaps surprising, the extent to which modern investigations into the molecular structure of the various channels have confirmed the physical reality, or approximate reality, of many aspects of the model.” Dr W.J. Heitler, University of St Andrews https://www.st-andrews.ac.uk/~wjh/hh_model_intro/
- 43. 43 Depolarization and hyperpolarization: the rabbits and foxes of a neuron The similarity between the HH model and the predator-prey model can be revealed by looking at the phase plane of the potassium gate (n) versus the voltage (V). For periodic spikes, we see a limit cycle. Izhikevich, 2007: Dynamical Systems in Neuroscience
- 44. 44 A single neuron as a dynamical system Izhikevich, 2007: Dynamical Systems in Neuroscience
- 45. 45 A single neuron as a dynamical system Izhikevich, 2007: Dynamical Systems in Neuroscience
- 46. 46 A single neuron as a dynamical system Izhikevich, 2007: Dynamical Systems in Neuroscience
- 47. 47 “Paradoxical” perturbations revisited Periodic firing – or a limit cycle
- 48. 48 “Paradoxical” perturbations revisited A transient increase in input disrupted firing. What happened?
- 49. 49 Dynamical systems used for prediction John Rinzel and colleagues analyzed how fixed points and limit cycles change with changes in parameters (inputs and initial conditions). They used this to show that for some types of neurons, a transient increase in input can shut down repetitive firing (!). They also verified this experimentally. Izhikevich, 2007: Dynamical Systems in Neuroscience
- 50. 50 Bumping the neuron into rest Rinzel and colleagues showed that in a type 2 neuron, i.e., one with two stable states, rest and repetitive firing, it is possible to “bump” the system from one attractor to the other with a small increase in input. Izhikevich, 2007: Dynamical Systems in Neuroscience Firing Attractor Resting Attractor
- 51. 51 Dynamical systems all the way down (and up!) The Pyramidal neuron Interneuron Gamma (PING) model Borgers (2017). An Introduction to Modeling Neuronal Dynamics
- 52. 52 Behavior as an attractor John, Y. J., Zikopoulos, B., Bullock, D., & Barbas, H. (2018). Visual Attention Deficits in Schizophrenia Can Arise From Inhibitory Dysfunction in Thalamus or Cortex. Computational Psychiatry, 2, 223-257.
- 53. 53 How the leopard got its spots? Kondo, S., & Miura, T. (2010). Reaction-diffusion model as a framework for understanding biological pattern formation. Science, 329(5999), 1616-1620.
- 54. 54 A “landscape” of memories Thinking of memories in terms of dynamical systems can help us understand their context-sensitive nature. A good memory system needs “deep” and “well- separated” attractor “valleys”. A memory system with shallow valleys can be bumped into a nearby but incorrect valley. Valleys can’t be too separated… or it will be difficult to find the “sweet spot” for memory recall! http://www.scholarpedia.org/article/Attractor_network Stephen Grossberg and John Hopfield were pioneers of the use of dynamical systems to model memory and other neural processes.
- 55. 55 Memory “landscapes” versus memory “slots” The dynamical system perspective can help us see the limitations of the computer metaphor (which is the same as the older library metaphor).
- 56. 56 The persistence of memory Constantinidis, C., & Klingberg, T. (2016). The neuroscience of working memory capacity and training. Nature Reviews Neuroscience, 17(7), 438.
- 57. 57 Conclusion: Why Dynamical Systems? Dynamical systems methods allow us to model any kind of time-varying system. Quantitative methods can be used to gain qualitative understanding. The concepts of attractors, repellers, limit cycles etc. can help us think about seemingly paradoxical phemomena. For memory, the dynamical systems perspective provides a useful alternative to the computer metaphor.
- 60. 60 Matlab
- 61. 61 Octave
- 62. 62 Lotka-Volterra: Things to Try Try different values of alpha, beta, delta and gamma Save the file after any editing To run, press F5 What happens if x = gamma/delta and y = alpha/beta?
- 63. 63 Hodgkin-Huxley: Things to Try Change the input strengths Try the ramping input Reduce the duration of the simulation to see the spike shape more clearly Change the ion channel parameters