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Neural Basis of Cognition Lecture 7 Learning and Memory: Part III LTP/LTD • In the last lecture, we briefly explored a candidate biochemical pathway for the induction of LTP/LTD • Questions: – Can we model the induction of LTP/LTD? – Is this pathway plausible? i.e. if we model this pathway, will our model neuron exhibit LTP/LTD as observed in real neurons? – If a neural network is equipped with such rules, can it “learn”? LTP/LTD • Can we model the induction of LTP/LTD? Yes. • Doing so will require some background. A Mathematical Detour • Next, we are going to consider the HodgkinHuxley model of the neuron. In order to do that, we will first review several topics: – Ordinary differential equations – Equilibrium potential – Electrical circuits • These tools will give us the ability to understand the basics of the Hodgkin-Huxley model Differential Equations • Given a function f(x), the average change in the function over a time h starting at a point x is “rise over run,” that is, the change in f(x) divided by the change in x – [f(x+h)-f(x)]/h Differential Equations • As we decrease h more and more until it approaches 0, the secant line becomes a line tangent to f(x) at the point x. Differential Equations • If we do not choose the point x at which we want to find the tangent line of f(x) and leave x as a variable, we can find the general equation for the tangent line of f(x) at any point x; this equation is the derivative of f(x), denoted f’(x) or df(x)/dx Differential Equations • Example: f(x) = x^2 [f(x+h) – f(x)] / h = [(x+h)^2 – x^2] / h = [(x^2 + 2xh + h^2) – x^2] / h = (2xh + h^2) / h = 2x + h lim (h->0) 2x+h = 2x So, the derivative of x^2 is 2x: f’(x) = 2x Differential Equations • What does the derivative mean? The derivative of a function describes how quickly the function changes with time. • If f(x) = x^2, f’(x) = 2x; this means that at any point x, f(x) is increasing by an amount equal to twice that x • Therefore, if we know that some quantity x increases twice as fast as its current value x, we know that the function that can give the value of that quantity at any point in time has derivative 2x, and therefore the function describing the change in that quantity over time is x^2. Differential Equations • The process of finding a function from its derivative is called antidifferentiation, or, more commonly, integration. • When describing physical systems, it is often easier to observe how a system changes based on its current state and to write that down an equation, called a differential equation; integrating this equation will lead to an equation that can describe the state of the system at any point in time. Equilibrium potential • The concentrations of an ion inside and outside of a cell are maintained by a combination of electrical and chemical gradients • The Nernst equation can be used to calculate the potential of an ion of charge z across a membrane. This potential is determined using the concentration of the ion both inside and outside the cell when it is at rest: – E = (RT/zF) ln([outside]/[inside]) • • • • R = universal gas constant, 8.314 Joules / (K mol) T = temperature in Kelvin z = charge of ion F = Faraday’s constant, 96485 Coulombs/mol • When the membrane potential is equal to E, there is no net flux of this ion across the membrane Equilibrium Potential • Using the Nernst equation, we can calculate the equilibrium potentials of Na+ and K+ in a neuron: – If, at resting potential, K+ has concentrations 5mM outside and 140 mM inside the cell, • EK = RT/F*ln(5/140) = -0.089 V = -89 mV – If, at resting potential, Na+ has concentrations 12 mM inside and 140mM outside the cell, • ENa = RT/F*ln(140/12) = 0.066 V = 66 mV Electrical Circuits • An electrical circuit is a closed loop through which current (electricity) runs. – “The voltage between two points is a short name for the electrical force that would drive an electric current between those points.” – “Electric current means, depending on the context, a flow of electric charge (a phenomenon) or the rate of flow of electric charge (a quantity). This flowing electric charge is typically carried by moving electrons, in a conductor such as wire….” Electrical Circuits • “A resistor is a two-terminal electronic component that produces a voltage across its terminals that is proportional to the electric current passing through it.” • “A capacitor is a passive electronic component consisting of a pair of conductors separated by a dielectric (insulator). When a voltage exists across the conductors, an electric field is present in the dielectric. This field stores energy.” – “An ideal capacitor is wholly characterized by a constant capacitance C, defined as the ratio of charge ±Q on each conductor to the voltage V between them: C = Q/V” Hodgkin-Huxley Model In 1952, Hodgkin, Huxley, and Katz, based on their experimental work with the squid giant axon, published their model describing how action potentials in neurons are initiated and propagated. They received the 1963 Nobel prize in Physiology or Medicine for their work. Their model has since been significantly expanded. The Hodgkin-Huxley model is based on an “equivalent circuit” for the neuron. The membrane is modeled as a capacitor, the voltage-gated channels as resistors, and the electrochemical gradient across the membrane for each ion as voltage sources. Hodgkin-Huxley Model • The current through each of the resistors (i.e. the current through each of the types of channels) is proportional to the difference between the voltage drop across the membrane and the relevant ion’s reversal potential, (Vm – Ek), where k is the channel type: ik = -gk(Vm-Ek), so Im = -Σk gk(Vm-Ek) • The proportionality factor gk is the conductance of the channel (the conductance of the resistor in the circuit). However, the conductance is generally not a constant and instead changes nonlinearly with the potential of the neuron. Hodgkin-Huxley Model • • • Recall that the capacitance of a capacitor (which is a constant) is described by the equation C = Q/V, rearrange, and differentiate with respect to time: CmVm = Q Cm dVm/dt = dQ/dt and the change in charge over time is current, so Cm dVm/dt = Im and Im was defined above, so Cm dVm/dt = -Σk gk(Vm-Ek) This is the fundamental equation describing the Hodgkin-Huxley equivalent circuit. Further complexity arises from the fact that gk is also dependent on voltage, which can make this differential equation more difficult to solve. LTP/LTD • Is this pathway plausible? Yes. LTP/LTD • “Calcium Time Course as a Signal for Spike-Timing–Dependent Plasticity” by Rubin et al. 2005 – Two-compartment model, for the dendrite and the soma – “The mechanism for plasticity in our model involves a biophysically plausible calcium detection system that responds to calcium and then changes the strength of the synapse accordingly. In the model, three detector agents (P, A, V) respond to the instantaneous calcium level in the spine compartment. The interactions of these three agents, together with two others (D, B) that they influence, act to track the calcium time course in the spine (see Fig. 3A). More specifically, different calcium time courses lead to different time courses of P and D, which compete to influence a plasticity variable W. This variable W is used as a measure of the sign and magnitude of synaptic strength changes from baseline. Note that this scheme is significantly different from a detection of peak calcium levels, in which the change in W would be determined by how large spine calcium becomes during an appropriate set of spikes. The interactions between agents within our detector system qualitatively resemble the pathways influencing the regulation of Ca2/calmodulin-dependent protein kinase II (CaMKII).” Rubin, J. E., R. C. Gerkin, et al. (2005). "Calcium time course as a signal for spike-timing-dependent plasticity." J Neurophysiol 93(5): 2600-13. LTP/LTD “Calcium has been proposed as a postsynaptic signal underlying synaptic spiketiming-dependent plasticity (STDP). We examine this hypothesis with computational modeling based on experimental results from hippocampal cultures… in which pairs and triplets of pre- and postsynaptic spikes induce potentiation and depression in a temporally asymmetric way. Specifically, we present a set of model biochemical detectors, based on plausible molecular pathways, which make direct use of the time course of the calcium signal to reproduce these experimental STDP results…. Simulations of our model are also shown to be consistent with classical LTP and LTD induced by several presynaptic stimulation paradigms.” Rubin, J. E., R. C. Gerkin, et al. (2005). "Calcium time course as a signal for spike-timing-dependent plasticity." J Neurophysiol 93(5): 2600-13. LTP/LTD • If a neural network is equipped with such rules, can it “learn”? Yes. LTP/LTD • “Learning Real-World Stimuli in a Neural Network with Spike-Driven Synaptic Dynamics” by Brader, Senn, Fusi 2007 – “We present a model of spike-driven synaptic plasticity inspired by experimental observations and motivated by the desire to build an electronic hardware device that can learn to classify complex stimuli in a semisupervised fashion. During training, patterns of activity are sequentially imposed on the input neurons, and an additional instructor signal drives the output neurons toward the desired activity. The network is made of integrate-and-fire neurons with constant leak and a floor. The synapses are bistable, and they are modified by the arrival of presynaptic spikes. The sign of the change is determined by both the depolarization and the state of a variable that integrates the postsynaptic action potentials. Following the training phase, the instructor signal is removed, and the output neurons are driven purely by the activity of the input neurons weighted by the plastic synapses. In the absence of stimulation, the synapses preserve their internal state indefinitely.Memories are also very robust to the disruptive action of spontaneous activity. A network of 2000 input neurons is shown to be able to classify correctly a large number (thousands) of highly overlapping patterns (300 classes of preprocessed Latex characters, 30 patterns per class, and a subset of the NIST characters data set) and to generalize with performances that are better than or comparable to those of artificial neural networks. Finally we show that the synaptic dynamics is compatible with many of the experimental observations on the induction of long-term modifications (spike-timingdependent plasticity and its dependence on both the postsynaptic depolarization and the frequency of pre- and postsynaptic neurons).” Brader, J. M., W. Senn, et al. (2007). "Learning real-world stimuli in a neural network with spike-driven synaptic dynamics." Neural Comput 19(11): 2881-912. LTP/LTD “A network of 2000 input neurons is shown to be able to classify correctly a large number (thousands) of highly overlapping patterns (300 classes of preprocessed Latex characters, 30 patterns per class, and a subset of the NIST characters data set) and to generalize with performances that are better than or comparable to those of artificial neural networks.” Brader, J. M., W. Senn, et al. (2007). "Learning real-world stimuli in a neural network with spike-driven synaptic dynamics." Neural Comput 19(11): 2881-912. LTP/LTD “Figure 6: The full Latex data set containing 293 classes. (a) Percentage of nonclassified patterns in the training set as a function of the number of classes for different numbers of output units per class. Results are shown for 1(∗), 2, 5, 10, and 40(+) outputs per class using the abstract rule (points) and for 1, 2, and 5 outputs per class using the spike-driven network (squares, triangles, and circles, respectively). In all cases, the percentage of misclassified patterns is less than 0.1%. (b) Percentage of nonclassified patterns as a function of the number of output units per class for different numbers of classes (abstract rule). Note the logarithmic scale. (c) Percentage of nonclassified patterns as a function of number of classes for generalization on a test set (abstract rule). (d) Same as for c but showing percentage of misclassified patterns.” Brader, J. M., W. Senn, et al. (2007). "Learning real-world stimuli in a neural network with spike-driven synaptic dynamics." Neural Comput 19(11): 2881-912.