Computational Neuroscience. Session 1-1roquesol/Computational_Neuroscience… · Neuroscience, The Geometry of Excitability and Bursting . 2007 MIT Press. Dayan Peter, Abbot L F

The CourseIntroduction

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Computational Neuroscience. Session 1-1

Dr. Marco A Roque Sol

05/29/2018

Dr. Marco A Roque Sol Computational Neuroscience. Session 1-1


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Course objectivesCourse outcomesTextbooks

Course objectives

Students will gain understanding of cell physiologyunderlying neuronal excitability.

Students will learn the Hodgkin-Huxley model of actionpotential generation and propagation.

Students will learn models of neuronal spiking and burstingof different levels of complexity.

Students will learn how to qualitatively analyze thebehavior of solutions of ordinary differential equationsusing phase-plane analysis.



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Course objectives

Students will understand the concept of bifurcation of adynamical system, and use it to analyze models ofneuronal excitability.

Students will learn how to use MATLAB to graphicallyanalyze and numerically solve ordinary differentialequations arising in neuronal modeling.

Students can describe physiological mechanismsunderlying an action potential in an excitable cell.



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Course outcomes

Students are able to analyze the behavior of a non-linearordinary differential equation using phase-plane analysis.

Students are able to build and analyze models of spikingand bursting neurons.

Students are able to write a MATLAB program tonumerically solve ordinary differential equations arising inthe modeling of neural excitability.

Students will learn how to qualitatively analyze thebehavior of solutions of ordinary differential equationsusing phase-plane analysis.



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Textbooks

Ermentrout G Bard, Terman David H . MathematicalFoundations of Neuroscience. 2010 Springer Science.

Izhikevich Eugene M. Dynamical Systems inNeuroscience, The Geometry of Excitability and Bursting .2007 MIT Press.

Dayan Peter, Abbot L F . Theoretical Neuroscience,Computational and Mathematical Modeling of NeuralSystem. 2001 MIT Press.

Squirre Larry S, Berg Darwing, Bloom Floyd E, Du LacSascha, Gosh Arnivan . Fundamental Neuroscience. 4thEdition, 2012 Academic Press.



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Textbooks

Johnston Daniel, Miao-Sin Wu Samuel . Foundations ofCellular Neurophysiology, with illustrations and simulationsby Richard Gray. 1995 MIT Press.

Wallisch Pascal, Lusignan Michael E, Benayoun Marc D,Baker Tanya I, Dickey Adam Seth, Hatsopoulos NicholasG. MATLAB for Neuroscientists, An introduction toScientific Computing in MATLAB. 2th Edition, 2014Academic Press.

Ermentrout G Bard. Simulating, Analyzing, and AnimatingDynamical Systems: A Guide to XPPAUT for Researchersand Students. 2012 SIAM.



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Table of ContentsIntroduction to Neuroscience.

Table of contents

1.- Introduction to Neuroscience.

2.- Basic introduction to the brain.

3- ODE Review .

4.-Intro to MATLAB.

5.- MATLAB and ODES’s

6.- The resting Potential.

7.- Nernst-Planck equation, Nernst equation, GHKequation. How to solve ODE.



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Table of contents

8.- Dynamics of Passive membrane.

9.- The Huxley-Hodgkin Model. Integrate-and-FireModels I.

10.-The Huxley-Hodgkin Model. Integrate-and-FireModels II.

11.- The Huxley-Hodgkin Model.The Cable equation I.

12.- The Huxley-Hodgkin Model. The Cable equation II



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Table of contents

13.- Introduction to Dynamical Systems for Neural Network.Reduced one and two-dimensional networks I (***).

14.- Introduction to Dynamical Systems for NeuralNetwork. Reduced one and two-dimensional networks II.

15.- One-dimensional Neural Model.Phase-Space Analysis I

∗ ∗ ∗ The idea in this part, in a regular class duringFall/Spring Semester, is to get support from either theDepartment of Biology or the Institute of Neurosciences, tosee some in vitro experiments.



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Table of contents

16.- Two-dimensional Neural Model.Phase-Space Analysis I

17.- Two-dimensional Neural Model.Phase-Space Analysis II

18.- Subthreshold Oscillations. Subthreshold andSuprathreshold Resonance



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Introduction to Neuroscience.

The field of Neuroscience had a breakthrough in 1952 when A.F. Huxley and A. L. Hodgkin published its paper where theydescribed, through a set of nonlinear Partial DifferentialEquations, the fundamental role of the action potential in theaxon of a giant squid.

Since that moment a great number of mathematicians havecontributed to the understanding of the field.

At the same time the growth of computer sciences has allowedto have a different perspective of the problem.



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Thus, using mathematical analysis and computer modeling, thearea of brain research has two powerful tools to study anddetermine how the nervous systems works.

In this course we will have the opportunity to realize howdifferent areas of the knowledge interact to give an explanationto physical phenomena.

In this way, we will make use of the concepts in Biology,Chemistry, Physics, and Physiology to generate Mathematicalmodels to understand the brain behavior.



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At the beginning ...

The Greek physician Galen thought that the brain was agland, from which nerves carry fluid to the extremities. Inthe mid-nineteenth century, it was discovered that electricalactivity in a nerve has predictable effects on neighboringneurons.

Camillo Golgi and Santiago Ramon y Cajal made the firstdetailed descriptions of nerve cells in the late nineteenthcentury, and one of Cajal’s drawings is shwon below:



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Ramon y Cajal drawing of a single neuron



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At the beginning ...Ross Harrison discovered that the axon and dendritesgrow from the cell body in isolated culture [1]; also see [2].

Pharmacologists discovered that drugs affect cells bybinding to receptors, which opened the door to thediscovery of chemical communication between neurons atsynapses.



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Over the past century neuroscience has now grown into abroad and diverse field. Molecular neuroscience studiesthe detailed structure and dynamics of neurons, synapsesand small networks;

systems neuroscience studies larger-scale networks thatperform tasks and interact with other such networks (orbrain areas) to form pathways for higher-level functions;

and cognitive neuroscience studies the relationshipbetween the underlying physiology (neural substrates) andbehavior, thought, and cognition.



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Mathematical treatments of the nervous system began inthe mid 20th century. One of the first examples is the bookof Norbert Wiener, based on work done with the Mexicanphysiologist Arturo Rosenblueth, and originally publishedin 1948 [3].

Rosenbleuth and WienerDr. Marco A Roque Sol Computational Neuroscience. Session 1-1


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Weiner introduced ideas from dissipative dynamicalsystems, symmetry groups, statistical mechanics, timeseries analysis, information theory and feedback control.He also discussed the relationship between digitalcomputers (then in their infancy) and neural circuits, atheme that John von Neumann subsequently addressed ina book written in 1955-57 and published in the year afterhis death [4].



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In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



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These books [3],[4] were directed at the brain and nervoussystem in toto, although much of the former was based ondetailed experimental studies of heart and leg muscles inanimals.

The first cellular-level mathematical model of a singleneuron was developed in the early 1950’s by the Britishphysiologists Alan Hodgkin and Andrew Huxley [5].



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This work, which won them the Nobel Prize in Physiologyin 1963, grew out of a long series of experiments on thegiant axon of the squid Loligo by themselves and others (see Huxley’s obituary [6] )

Since their pioneering work, mathematical neurosciencehas grown i nto a subdiscipline, served worldwide bycourses short and long, a growing list of textbooks (e.g. [7],[8], [9], [10],[11], [12]) and review articles such as [13],[14], [15], [16].



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Mathematical Neuroscience

Mathematical Neuroscience here means an area ofneuroscience where mathematics is the primary tool forelucidating the fundamental mechanisms responsible forexperimentally observed behaviour.

Drawing together, the field provides the possibility of acritical discussion of the relevant experimental facts and ofvarious mathematical methods and techniques that havebeen successfully applied to date.



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More importantly, it can draw attention to, and develop,those pieces of mathematical theory which are likely to berelevant to future studies of the brain [17]. In illustration ofthis point it is worth telling the story of Wilfrid Rall, who inthe 1960s developed the cable model of the dendritic tree(see [18] for a survey of his work)

Cable theory uses coupled PDEs (Partial Differential;Equations) to describe how membrane potential spreadsalong the dendritic branches in response to a localconductance change (synaptic input).



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Using his mathematical formalism, Rall showed that thereis a subclass of trees that is electrically equivalent to asingle cylinder whose diameter is that of the stem dendrite.

To a first approximation, many neurons (e.g.α-motoneuron) belong to this subclass (though cortical andhippocampal pyramidal cells do not). Importantly Rall’s“equivalent cylinder” model allows for a simple analyticalsolution and this has provided the main insights regardingthe spread of electrical signals in passive dendritic trees.



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As another example we turn to work on neural fieldequations in the 1970’s, by people such as Hugh Wilson,Jack Cowan, Bard Ermentrout, Shun-ichi Amari, PaulNunez and Hermann Haken (for a recent overview see[19])

These are tissue level models that describe thespatio-temporal evolution of coarse-grained variables suchas synaptic or firing rate activity in populations of neurons,and often take the form of integro-differential equations.



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The sorts of dynamic behaviour that are typically observedin neural field models include spatially and temporallyperiodic patterns (beyond a Turing instability[morphogenesis]), localised regions of activity andtravelling waves.

The mathematical study of such equations and theirsolutions has proven relevant to understanding EEGrhythms( Electroencephalography), mechanisms forshort-term memory, motion perception and drug-inducedvisual hallucinations.



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In this latter context the use of symmetric bifurcation theoryhas shown that neural activity patterns underlying commonvisual hallucinations can be accounted for in terms ofcertain symmetry properties of the anisotropicsynapticconnections in visual cortex (requiring the use of anovel representation of the planar Euclidean group) [20]

As well as the above examples of the practice ofmathematical neuroscience, it is as well to mention someof the tools in the arsenal of the mathematicalneuroscientist world.



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It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date. Indeed, seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved on to encompass increasinglysophisticated tools of modern applied mathematics (Topology, Algrebraic Geometry, Category Theory ).



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Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



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The field is now in the healthy state where not only ismathematics having an impact on neuroscience, the latteris simultaneously motivating important research inmathematics. In recent years a number of high profilemathematical institutes, including the MathematicalSciences Research Institute (Berkeley; 2004), theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and the Centre de Recerca Matematica(Andorra; 2006), have held workshops with the title“Mathematical Neuroscience“ (of at least three daysduration)



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One area in which neuroscience has already prompted thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



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Mathematical Neuroscience workshop to be held at theCentre de Recherches Mathematiques, Universite deMontreal in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony” is agood example of the relevance of mathematics inneuroscience, where it is perhaps more well known for itsdiscussion in relation to the binding problem [30] and brainrhythms in general [31].



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Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



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Basic introduction to the brain.References

Basic introduction to the brain

Motivation. This lectures are intended to give students ageneral intuition for basic mathematical language used todescribe and model neurons. These principles will serveas the foundation for future sections.



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Basic introduction to the brain.

Basic organization of the brain. The brain is typicallydivided into 4 lobes. The temporal lobe contains neuralmachinery for processing speech and sounds, spatialinformation, and for encoding episodic (autobiographical)memories. The parietal lobe is involved with sensoryperception, sensory integration, and memory.



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Basic organization of the brain.The frontal lobe is associated with personality, reasoning,planning, problem solving, working memory, andmovement. The occipital lobe is primiarily involved withvision and visual processing. The central sulcus separatesthe primary motor cortex (frontal lobe) from the primarysomatosensory cortex (parietal lobe).



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Basic organization of the brain. The medial longitudinalfissure’ separates the right and left hemispheres of thebrain. The spinal cord sends signals from the primarymotor cortex to the body’s skeletal muscles.



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Neuron anatomy. Neurons are electrically excitable cellsin the brain. Neurons “listen“ to other cells via branch-likedendrites. Signals from the dendrites travel down to thecell body, or soma, which contains the nucleus of the cell.Neurons communicate with other cells by sendingelectrical impulses down their axons, which most oftensynapse onto the dendrites of other neurons.



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Neuron anatomy. There are only around 1011 neurons inthe human brain, much fewer than the number ofnon-neural cells such as glia.

Yet neurons are unique in the sense that only they cantransmit electrical signals over long distances.

From neuronal level we can go down to cell biophysics, tomolecular biology of gene regulation, etc. From neuronallevel we can go up to neuronal circuits, to corticalstructures, to the whole brain, and finally to the behavior ofthe organism



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Action potentials. Neurons communicate w ith each otherby changes in their membrane voltage (we’ll get to whatthis means in a bit). Small changes in membrane voltageare picked up by neurons up to approximately 1mm away.Thus, for nervous systems on the scale of 1mm, such asthe fruit fly nervous system, no other special mode ofcommunication is needed.



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Action potentials. However, in larger nervous systems(e.g. ours), neurons fire action potentials - suddenchanges in voltage. These form the basic mode of neuralcommunication in the brain. Over the next few lectureswe’ll be trying to understand how action potentials comeabout by modeling neurons in increasing levels of detail.



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What is a spike?.

A typical neuron receives inputs from more than 10,000other neurons through the contacts on its dendritic treecalled synapses:



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What is a spike?.

The inputs produce electrical transmembrane currents thatchange the membrane potential of the neuron. Smallsynaptic currents produce small changes, calledpost-synaptic potentials (PSPs).

Larger currents produce significant PSPs that could beamplified by the voltage- sensitive channels embedded inneuronal membrane and lead to the generation of anaction potential or spike - an abrupt and transientchange of membrane voltage that propagates to otherneurons via a long protrusion called an axon.



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What is a spike?.

Such spikes are the main means of communicationbetween neurons. In general, neurons do not fire on theirown, they get fired by the incoming spikes from otherneurons. One of the most fundamental question ofneuroscience is: what exactly makes neurons fire?

What is it in the incoming pulses that elicits a response inone neuron but not in another one? Why could twoneurons have different responses to exactly the same inputand identical responses to completely different inputs?



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What is a spike?.

To answer these questions, we need to understand thedynamics of spike-generation mechanisms of neurons.

Most introductory neuroscience books describe neuronsas integrators with a threshold: Neurons sum up incomingPSPs and ”compare” the integrated PSP with a certainvoltage value, called firing threshold.



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What is a spike?.

If it is below the threshold, the neuron remains quiescent;when it is above the threshold, the neuron fires anall-or-none spike and resets its membrane potential.

To add theoretical plausibility to this argument, we have torefer to the Hodgkin-Huxley model of spike-generation insquid giant axons.



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The neuron as a fluid-filled ball. Each 3µm of cytoplasmcontains on the order of 1010 water molecules, 108 ions(e.g. sodium, potassium, calcium, chloride), 107 smallmolecules (e.g. amino acids, nucleicacids), and 105

proteins. Relative to the extracellular space, the inside ofthe cell is negatively charged (the difference is carried byabout 1 out of every 100,000 ions). This results in avoltage (V ) across themembrane of approximately−70mV .



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The neuron as a capacitor. Excess negative charges inthe cell oppose each other and line up around inside ofmembrane. This attracts an equal number of extracellularpositive ions, which line up outside the cell. In this way, themembrane builds up charge - it’s acting as a capacitor!The amount of charge (Q) stored by the membrane isgiven by the following equation: CmV = Q



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That is, the amount of charge stored by the membrane isequal to the ability of the membrane to store charge (i.e.,its capacitance) multiplied by the voltage difference acrossthe membrane. The total membrane capacitance (Cm ) isproportional to the surface area of the cell (A):

Cm = cmA



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Specific capacitance (cm) depends on conductance andthickness of membrane, which is about the same for allneurons - about 10nF /mm2. Neurons typically have asurface area of 0.01− 0.1mm2 , so Cm ranges from around0.1− 1nF . We can now compute the number of chargesstored by a given neuron (we’ll assume 1nF totalcapacitance and −70mV membrane potential):

1nF ×−70mV = 10−9F ×70×10−3V = 70×10−12C = 109charges.

Note: A Columb is 1 Farrad × volt.



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Changes in current. Membrane current is a measure ofthe number of charges per second that travel across themembrane. Current is measured in amps - 1amp is 1Columb per second:

I =dQdt

In order to compute the membrane current, we can takethe time derivative of the equation for determining howmuch charge the membrane stores:

CmdVdt

=dQdt

= I

Example: suppose Cm = 1nF . Then injecting I = 1nA ofcurrent causes the membrane voltage to rise by 1 volt persecond (i.e., 1mV per millisecond ).



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Membrane current. There are two components of current(I). The first is membrane current. The membrane containsion channels - these let specific neurons through. Theycan open and close. One type of channel is the sodiumchannel. The inside of the cell contains fewer sodium ionsthan outside the cell.



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When the sodium channels open, sodium (positivelycharged) flows into the cell and causes the membranevoltage to increase. Diffusion of sodium and other ions(e.g. potassium) is called the membrane current, Im .



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Driving force and the equilibrium potential. In additionto sodium being driven to flow down its concentrationgradient, one can make it more or less difficult for sodiumto enter the cell by changing the membrane voltage.Because sodium is positively charged, decreasing, orhyperpolarizing, the membrane voltage (inside relative tooutside) will make sodium ions more likely to flow into thecell.



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Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



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External current. The second component of current iscurrent that is injected into the neuron from externalsources (e.g. if we stick an electrode into the neuron andpump in current). The change in membrane voltage V dueto some change in current I follows Ohm’s Law: V = IR,where R is the membrane resistance, described next.

Membrane resistance. Ion channels are like little holes inthe membrane. They let ions pass through them - i.e., theyconduct ions. A given unit area of membrane has somenumber of open channels, and we can measure the easewith which ions pass through those channels - the specificconductance, gm. The total conducatance is proportionalto the neuron’s area: Gm = gmA



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By convention, we tend to talk about the inverse ofconductance, which is called resistance. Whereasconductance is proportional to the surface area of theneuron, resistance is proportional to the inverse of thesurface area of the neuron:

Rm =rm

A

. Note that the membrane resistance often changes as afunction of voltage, which makes things interesting.



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The Neuron Equation. Previously we had:

CmdVdt

= I

which we can update to reflect that I is comprised of bothmembrane and external currents:

CmdVdt

= Ie + Im



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Im depends on the driving force V − E and also difficultywith which ions flow through the membrane - i.e., themembrane resistance, Rm. In particular

Im =1

Rm(E − V )

The Neuron Equation. Note that the order of the E and Vterms in the driving force have been swapped. This isbecause the internal and external currents need to go inopposite directions. We can multiply both sides of theequation by Rm for convenience:

CmRmdVdt

= RmIe + (E − V )



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Since Cm = cmA and Rm = rmA , the A′s cancel, and we get

cmrm, which is independent of the cell’s surface area.Since cmrm determines the rate at which the cell’smembrane potential changes, it is given a special variable,τm, or the membrane time constant. The equation for allneuron models we’ll see in this class is:

τmdVdt

= RmIe + E − V



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V∞. From the above equation, we see that the change inmembrane voltage is some fraction (proportional to τm ) ofthe difference between RmIe + E and the currentmembrane voltage, V . By this equation the membranevoltage approaches RmIe + E over time. For conveniencewe can define

V∞ = RmIe + E

where V∞ is the membrane voltage that will be reachedgiven an external current and membrane resistance, andan infinite amount of time.



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Resting potential. If you shut off the external current (i.e.,set Ie = 0), then V∞ = E . For this reason, we call E theresting potential of the cell - the potential the cellapproaches if we remove external forces.



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Computing the voltage at a particular time, t. We needto solve the following differential equation for V :

τmdVdt

= RmIe + (E − V )

We know that, given enough time, V tends towards V∞ , sowe can say that at time t :

V (t) = V∞ + f (t)

Now we need to find f (t). We use the equation above,substituting in V∞ + f for V



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τmdfdt

= RmIe + E − V∞ − f

Since V∞ = RmIe + E , those terms cancel and we have:

τmdVdt

= −f

Integrating this differential equation we have

τmdfdt

= −f

τmdf = −fdt)



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∫ f (0)

f (t)τmdf = −

∫ t

0fdt

τm[lnf (t)− lnf (0)] = −t

τmln[f (t)f (0)

] = −t

ln[f (t)f (0)

] = − tτm



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f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm

Previously, we said that the voltage changes as a functionof the distance between the voltage at the present timeand V∞. So f (0) = V (0)− V∞ , and f (t) = f (0)e−

tτm .

Plugging f (t) back into the original equation gives:

V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm

This gives us a way to compute how long it will take tocharge up the neuron to an arbitrary voltage V (t), bysolving for t .



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References

[1] R. Harrison. The outgrowth of the nerve fiber as a mode ofprotoplasmic movement. J. Exp. Zool. , 9:787-846, 1910.

[2] H. Keshishian. Ross Harrison’s ’The Outgrowth of the NerveFiber as a Mode of Protoplasmic Movement’. J. Exp. Zool. ,301A:201-203, 2004.

[3] N. Wiener. Cybernetics: or Control and Communication inthe Animal and the Machine. M.I.T. Press, Cambridge, MA,1948. 2nd Edition, 1961.

[4] J. von Neumann. The Computer and the Brain . YaleUniversity Press, New Haven, CT, 1958. 2nd Edition, with aforeward by P.M. and P.S. Churchland, 2000.



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References

[5] A.L. Hodgkin and A.F. Huxley. A quantitative description ofmembrane current and its application to conduction andexcitation in nerve. J. Physiol., 117:500-544, 1952.

[6] M.C. Mackey and M. Santillan. Andrew Fielding Huxley(1917-1952). AMS Notices, 60 (5):576-584, 2013.

[7] H. Wilson. Spikes, Decisions and Actions: The DynamicalFoundations of Neuroscience. Oxford University Press, Oxford,U.K., 1999. Currently out of print, downloadable fromhttp://cvr.yorku.ca/webpages/wilson.htm# book.

[8] P. Dayan and L.F. Abbott. Theoretical Neuroscience:Computational and Mathematical Modeling of Neural Systems.MIT Press, Cambridge, MA, 2001.



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References

[9] E.M. Izhikevich. Dynamical systems in neuroscience: Thegeometry of excitab ility and bursting. MIT Press, Cambridge,MA, 2007.

[10] J. Keener and J. Sneyd. Mathematical Physiology.Springer, New York, 2009. 2nd Edition, 2 Vols

[11] G.B. Ermentrout and D. Terman. MathematicalFoundations of Neuroscience . Springer, New York, 2010.

[12] F. Gabbiani and S. Cox. Mathematics for Neuroscientists.Academic Press, San Diego, CA, 2010.



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References

[13] X.J. Wang. Neurophysiological and computationalprinciples of cortical rhythms in cognition. Physiol. Rev. ,90:1195-1268, 2010.

[14] N. Kopell. Toward a theory of modelling central patterngenerators. In A.H Cohen, S. Rossig- nol, and S. Grillner,editors, Neural Control of Rhythmic Movements in Vertebrates,pages 3369âAS413. Wiley, New York, 1988.

[15] M.M. McCarthy, S. Ching, M.A. Whittington, and N. Kopell.Dynamical changes in neurological disease andanesthesia.Curr. Opin. Neurobiol. , 22 (4):693-703, 2012.

[16] G. Deco, E.T. Rolls, L. Albantakis, and R. Romo. Brainmechanisms for perceptual and reward-relateddecision-making. Prog. in Neurobiol. , 103:194-213, 2013



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References

[17] J S Griffith. Mathematical Neurobiology: An introduction tothe mathematics of the nervous system. Academic Press, 1971.

[18] I Segev, J Rinzel, and G M Shepherd, editors. Thetheoretical foundations of dendritic function: selected papers ofWilfrid Rall with commentaries. MIT Press, 199.

[19] S Coombes. Waves, bumps, and patterns in neural fieldtheories. Biological Cybernetics, 93:91âAS108, 2005.

[20] C Bressloff, J D Cowan, M Golubitsky, P J Thomas, and MWiener. Geometric visual hallucinations, Euclidean symmetryand the functional architecture of striate cortex. PhilosophicalTransactions of the Royal Society London B, 40:299âAS330,2001.



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References

[21] S Coombes and M R Owen. Evans functions for integralneural field equations with Heaviside firing rate function. SIAMJournal on Applied Dynamical Systems , 34:574âAS600, 2004.

[22] P. Ashwin and M Timme. When instability makes sense.Nature, 436:36-37, 2005.

[23] J Rubin and D Terman. Handbook of Dynamical SystemsII, chapter Geometric Singular Perturbation Analysis ofNeuronal Dynamics. Elsevier, 2002.

[24] A Longtin and P Swain, editors. Stochastic Dynamics ofNeural and Genetic Networks . Special Focus Issue of CHAOS,Vol.16, 2006.



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References

[25] D Cai, L Tao, A V Rangan, and D W McLaughlin. Kinetictheory for neuronal network dynamics. Communications inMathematical Sciences, 4:97-127, 2006.

[26] P C Bressloff. Spatially periodic modulation of corticalpatterns by long-range horizontal connections. Physica D,185:131âAS157, 2003.

[27] Moehlis. Canards for a reduction of the Hodgkin-Huxleyequations. Journal of Mathematical Biology , 52:141âAS153,2006.

[28] S Ikeda, T Tanaka, and S Amari. Stochastic reasoning, freeenergy, and information geometry. Neural Computation,16:1779-1810, 2004.



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References

[29] M Rosenblum and A Pikovsky. Delayed feedback control ofcollective synchrony: An approach to suppression ofpathological brain rhythms. Physica Review E , 70(041904),2004.

[30] W Singer. Synchronization of cortical activity and itsputative role in information processing and learning. AnnualReview of Physiology, 55:349âAS374, 1993.

[31] G Buzsaki. Rhythms of the Brain. Oxford University Press,2006.

[32] D Robbe, S M Montgomery, A Thome, P E Rueda-Orozco,B L McNaughton, and G Buzs ÌAaki. Cannabinoids revealimportance of spike timing coordination in hippocampalfunction. Nature Neuroscience, 9:1526âAS1533, 2006



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References

[33] F Sargolini, M Fyhn, T Hafting, B L McNaughton, M PWitter, M-B Moser, and E I Moser. Conjunctive representationof position, direction, and velocity in entorhinal cortex. Science,312:758âAS762, 2006.


Documents

Computational Neuroscience. Session 1-1roquesol/Computational_Neuroscience… · Neuroscience, The Geometry of Excitability and Bursting . 2007 MIT Press. Dayan Peter, Abbot L F