SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**

Consortium leader

PETER PAZMANY CATHOLIC UNIVERSITY

Consortium members

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***

**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben

***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.

PETER PAZMANY CATHOLIC UNIVERSITY

SEMMELWEIS UNIVERSITY

Peter Pazmany Catholic University Faculty of Information Technology

MODELLING NEURONS AND NETWORKS

Lecture 4

Analysing neurons as dynamical systems

www.itk.ppke.hu

(Idegsejtek és neuronhálózatok modellezése)

( Neuronok, mint dinamikus rendszerek analízise )

Szabolcs Káli

Overview

In this lesson you will learn about:

• One-dimensional, nonlinear neuron models

• Stability analysis, equilibrium points

• Phase portraits

• Bifurcation diagrams

• 2-D systems, vector fields, limit cycles

• Types of bifurcation: Saddle-node, Andronov-Hopf

• Bifurcations of stable equilibria

• Bifurcations of stable limit cycles

• Hodgkin’s classification of neurons

• Integrator and resonator type neurons

• Spiking threshold analysis

The neuron membrane as a dynamical system

The simplest active membrane model with one gating variable (p):

Where V is the time-dependent membrane potential, the capacitance C, leak conductance g_l, and leak reverse potential E_l are constant

parameters; is the steady-state activation function:

If then the gating process can be treated as being

One dimensional, nonlinear model

A phenomenological model of the voltage-dependent membrane time constant is:

instantaneous:

This reduces the two-dimensional system to a one-dimensional equation. Note: we will use this form throughout the lesson.

The basic types of one-dimensional models

Since the channel current can be either inward or outward, and the gating process can be either activation and inactivation, there are four different types of models (I_Na,p, I_k, I_h, I_Kir):

Left: Steady-state current-voltage relations for the four models.

The leak + fast I

current model

Left: the I-V curves of the Na current, the leak current and the sum of the two currents.

Right: The derivative of the membrane potential (which is –I(V)/C)

Due to the negative conductance region the model can exhibit various types of non-linear behavior, such as bistability.

The leak + fast I

current model

Depending on the injected current the model can be either bistable or monostable.

Left: The I-V curves with different starting membrane potentials.

Right: I-V curves with injected current. Because of the injected current the system becomes monostable.

Geometrical analysis

• If we plot the derivative of the membrane potential (F(V)), we can determine the stable points of the system.

• If F(V) is negative, the membrane

potential decreases, and if it is positive, the membrane potential increases.

• If F(V)=0 the system is in an equilibrium (fixed point), since the change of the membrane potential is 0 over time.

Figure: Stability analysis of the I_leak

model(left column) and the I_Na,p model (right column). The I_leak model has one fixed point, while the I_Na,p model has three.

The stability of the fixed point

In 1D systems:

•Stable fixed point (or stable equilibrium): If (the slope is negative) near the stable state, then the fixed point is a stable fixed point.

•Otherwise it is an unstable fixed point.

•Between two stable fixed points there is always an unstable fixed point, because F(V) has to change sign from “-” to “+”.

In higher dimensions, other types of attractors can exist:

in 2 or more dimensions, stable curves called limit cycles (see later);

The stability of the fixed point Figure: mechanistic

interpretation of the phase space.

The massless ball always moves downwards with speed

proportional to the slope.

Unstable fixed points correspond

to local maxima, stable points

correspond to local minima.

The stability of the fixed point – attraction domains

Figure: An unstable equilibrium separates the attraction

domains. The direction of the slope is the direction of the membrane potential change.

The slope can be viewed as the

total membrane current.

Phase-space analysis of action potential generation Figure: The state changes

during action potential generation.

The subthreshold perturbation is caused by some external force (e.g.: excitatory postsynaptic potentials, injected current,…).

Repolarization is caused by another variable (K current)

which activates if the membrane potential is high enough.

Bottom figure: Upstroke

dynamics from in vivo

measurements.

Bistability and hysteresis

Figure: Bistability in a cat thalamocortical neuron.

After an action potential the cell will be in a stable state. A

hyperpolarizing current switches the system into another stable

state. The failure of the system to return to the original value

when the injected current is removed is called hysteresis.

Bistability and hysteresis

Figure: The injected current can shift the I-V curve up and down, causing stable points to appear or disappear.

Left: Different amounts of constant current injection change the number of stable points.

Right: If I were a slow V-dependent variable, then the system

could exhibit relaxation oscillations.

Phase portrait

Construction of the phase portrait: From the F(V) function we

can determine the attraction domains. By mapping multiple

attraction domains onto one line, we get the phase portrait of

the system.

Phase portrait – topological equivalence

Two phase portraits are topologically equivalent if they have the same type and number of equilibrium points in the same order.

Left: Two seemingly different systems which are topologically equivalent.

Right: Two similar-looking systems that are not topologically

equivalent.

Topological equivalence

The nonlinear system is locally (topologically) equivalent near the equilibrium point V=V_eq with the linear system if

(Hartman-Grobman theorem)

Bifurcation

A system is said to undergo a

bifurcation when its phase portrait changes qualitatively.

Pictures: Example of bifurcation.

The energy landscape changes, so

that the bistable system becomes

monostable.

Bifurcation is a qualitative change in the phase portrait, and it is not equivalent to a qualitative change of behavior.

change of behavior but no bifurcation

Bifurcation

Electrophysiological example of bifurcation

Pictures: qualitative change of behavior in a rat pyramidal cell.

By injecting current into the soma the cell becomes monostable.

Bifurcation in the I

model

The qualitative behaviour of the model depends on whether I is greater or less than 16. When I=0 (top figure), the system is bistable. The rest and the excited states coexist. When I is large (bottom figure) the rest state no longer exists because leak outward current cannot cope with large injected dc-current I and the inward Na⁺ current.

saddle-

node not saddle-node

Saddle-node bifurcation

is a 1-D system, with I=I_sn parameter and V=V_sn fixed point; it is at a saddle-node bifurcation if

Where

Slow transition near bifurcation

Slow transition through the ruins of the resting state attractor in a cortical pyramidal neuron with I =30pA injection. Even though

the resting state has already disappeared, the function F(V), and hence the rate of change in V, is still small when V is approximately -40 mV.

Slow transition near bifurcation

The voltage trace of the same neuron as in the previous slide. The attractor ruins (part of the phase space where the stable point

disappeared) are outlined in yellow.

Bifurcation diagram

To make the bifurcation diagram, we determine the locations of the stable and unstable equilibria for each value of the parameter I and plot them in the (I; V) plane. These points form two curves, which join at the saddle-node bifurcation. Unstable equilibrium points are denoted by open circles. As the bifurcation parameter I grows, the stable and

Steady-state I-V curve and bifurcation diagram

To find equilibrium points we plot the steady-state I-V curve I_∞(V ) and draw a horizontal line with altitude I (left figure). Any

intersection is an equilibrium and satisfies the equation I = I_∞(V ).

When I increases past 16, the saddle-node bifurcation occurs.

If we rotate this diagram by 90 degrees, we get an illustration where every vertical slice represents a specific phase portrait for I=const (right figure).

Another bifurcation in the l

model

If we inject negative current into the neuron another bifurcation appears:

The excited state disappears at sufficiently high negative current injection, because I becomes so negative that the Na⁺ inward current is no longer enough to balance the leak outward current and the negative injected dc-current to keep the membrane in the depolarized (excited)

state.

2-D (planar) systems

Most concepts will be illustrated using the I_Na,p+I_K model:

leak I_L instantaneous I_Na,p I_K

The state of the model is a two-dimensional vector (V;n) in the phase plane . Many interesting phenomena can be analysed using two-dimensional systems, such as neuronal bursting.

The newly introduced I_K current is relatively slow compared to the Na current, and has a low reversal potential.

2-D (planar) systems – I

current

The relatively slower persistent K⁺ current I_K has either high (left) or low (right) threshold. The two choices result in fundamentally different dynamics.

Each point in the phase plane (x;y) has its own vector (f;g), the system is said to define a vector field in the plane, also known as a direction field or velocity field. Thus, the vector field defines the direction of motion in each coordinate.

Nullclines are parts of the vector fields where one derivative is null:

Vector fields, nullclines, trajectories

which are called x-nullclines and y-nullclines respectively.

Solutions are trajectories tangent to the vector field.

Two-dimensional systems are often written in the form:

Vector fields, nullclines

Figure: Nullclines of the I_Na+I_K model. Nullclines are denoted by bold dashed and solid lines. Dotted line shows a possible spike trajectory.

Limit cycles

Stability of equilibria

Neutrally stable (not asymptotically stable) equilibria:

Some trajectories neither converge to nor diverge from the equilibria.

Unstable equilibria: An equilibrium is unstable if it is not stable.

Stability of equilibria

Consider the two-dimensional dynamical system

with an equilibrium point at (x₀,y₀). The partial derivatives evaluated at the equilibrium are:

Thus the Jacobian matrix of the system in this equilibrium point is:

Classification of equilibria

Classification of equilibria according to the trace ( ) and the determinant ( ) of the Jacobian matrix L. Shaded region

Classification of equilibria

The equilibria can be classified by looking at the eigenvalues of the 2D system

equation at the equilibrium point:

Stable node: The eigenvalues are negative and real numbers.

Unstable node: The eigenvalues are positive and real numbers.

Saddle: The eigenvalues are real and have opposite sign.

Unstable focus: The eigenvalues are

complex-conjugate and have positive real parts.

Stable focus: The eigenvalues are

complex-conjugate and have negative real parts.

Phase portrait of the I

+I

(high-threshold) model

Bistability

The attraction domains are separated by a pair of

trajectories, called separatrices, that converge to the saddle

equilibrium. Such trajectories

form a stable manifold of a saddle point.

The unstable manifold of a saddle is formed by the two trajectories that originate exactly from the saddle.

Picture: A brief current pulse

(arrow) brings the state vector of the system into the attraction domain of the stable limit cycle.

Picture: Stable and unstable manifolds to a saddle.

Homoclinic/heteroclinic trajectories: Heteroclinic orbit starts and ends at different equilibria. Homoclinic orbit starts and ends at the same equilibrium.

Bistability

Amplifying and resonant gates/currents

currents

Table: Amplifying variables amplify voltage changes trough a positive feedback loop (For example the activation variable m for inward, or the inactivation variable h for outward currents).

Resonant variables counter voltage changes via a negative feedback loop (e.g., causing depolarization when the cell is hyperpolarised and causing hyperpolarisation when the cell is depolarised.).

Minimal spiking models

Any combination of an amplifying and a resonant gating variable results in a spiking model.

Saddle-node bifurcation (1)

In the current state of the depicted system there is a stable node and an unstable node, connected by two heteroclinic orbits (since

trajectories do not end at the same equilibrium states (nodes)).

Saddle-node bifurcation (2)

As the injected current increases, the two nodes merge, and a homoclinic orbit is formed.

Saddle-node bifurcation (3)

Further increasing the current makes the node disappear, resulting in a limit cycle attractor.

Saddle-node bifurcation

As the injected current slowly increases, the system moves from one equilibrium state to a limit cycle without equilibrium points.

Top picture: Membrane voltage trace during increasing current injection.

Bottom picture: The I-V diagram of the model. Injected current can be visualised as a vertical line, which moves to the right as it increases

Saddle-node bifurcation

Picture: Saddle-node

bifurcation visualised in vector fields.

The unstable saddle and the stable node approach each other, coalesce, and terminate each other. Saddle-node

bifurcation occurs, when the saddle and the node are on the same spot in the vector field.

Yellow area denotes the basin of attraction of the stable

node.

Saddle-node on invariant circle bifurcation

(b) saddle-node on invariant circle (SNIC) bifurcation

Picture: Two types of saddle-node bifurcation. In the case of the saddle-node bifurcation the limit cycle does not include the node and the saddle. In the case of

Saddle-node on invariant circle bifurcation

Picture: Two types of saddle-node bifurcation visualised with vector fields in the I_Na,p+I_K model. Depending on the time constant of the K current a saddle- node (top picture) or an saddle-node on invariant circle bifurcation (SNIC) (bottom picture) occurs.

Bifurcation and membrane potential

Saddle-node on invariant circle bifurcation causes the cell to be able to spike at arbitrarily small frequencies.

The trajectory moves fast from B to A (spiking) and slowly from A to B (yellow region, interspike interval)

Andronov-Hopf bifurcations

In some cases neurons can exhibit another type (not saddle-node) bifurcation: the cell switches slowly from the non-spiking state to the spiking state. This type is called an Andronov-Hopf bifurcation.

Supercritical Andronov-Hopf bifurcation

In the supercritical Andronov-Hopf bifurcation a stable focus becomes a limit cycle: As the injected current increases the stable

Supercritical Andronov-Hopf bifurcation

Another example of supercritical Andronov-Hopf bifurcation: As the injected current increases the spiking activity is blocked by strong excitation.

The unstable equilibrium (which is the intersection point of the nullclines) moves to the right of the V-nullcline and becomes stable as injected current increases (see next page). The limit cycle shrinks and spiking activity disappears.

Top picture: measurements from rat’s visual cortex.

Bottom picture: Bifurcation in the I_Na,p+I_K model

Supercritical Andronov-Hopf bifurcation

Pictures: The disappearance of the limit cycle as the injected current

Top left: initial state, one stable equilibrium.

Top right: State after bifurcation, the middle region loses stability and gives birth to a stable limit cycle.

Right: Snapshots of the phase space with increasing currents.

Supercritical Andronov-Hopf bifurcation

subcritical Andronov-Hopf bifurcation

Subcritical Hopf bifurcation occurs when an unstable limit cycle shrinks to an equilibrium and makes it lose stability as the bifurcation parameter I increases.

Subcritical Andronov-Hopf bifurcation

Bifurcations of stable equilibria (summary)

Bifurcations of stable limit cycles

Bifurcations of stable limit cycles

I denotes the amplitude of the injected current, I_b is the bifurcation value, A is a parameter that depends on the biophysical details.

Neuronal excitability, bistability, oscillations

A dynamical system having a stable equilibrium is excitable if there is a large-amplitude trajectory that starts in a small neighborhood of the equilibrium (yellow), leaves the neighborhood, and then returns

Excitable dynamical systems can bifurcate into oscillatory systems

Figure: The change from a stable equilibrium to an oscillatory system.

Depending on the type of the bifurcation of the limit cycle, the equilibrium may disappear (saddle-node on invariant circle bifurcation) or may lose stability (supercritical Andronov-Hopf bifurcation).

Alternatively, the equilibrium may remain stable and co-exist with the newly appeared limit cycle, as it happens during saddle homoclinic orbit or fold limit cycle bifurcations. The dynamical system becomes bistable.

Since the equilibrium is near the cycle in a bistable system , a small modification of the vector field in the shaded neighborhood can make it disappear via saddle-node bifurcation, or lose stability via subcritical Andronov-Hopf bifurcation.

Hodgkin's classification

Hodgkin classified neurons into two categories (in 1948, years before mathematicians discovered such bifurcations):

• Class 1 neural excitability: Action potentials can be generated with arbitrarily low frequency,

depending on the strength of the applied current (left picture).

• Class 2 neural excitability: Action potentials are generated in a

certain frequency range that is relatively insensitive to changes in the strength of the applied current (right picture).

Class 3 excitability

Hodgkin also observed that axons left in oil or sea water for long

periods exhibited Class 3 neural excitability. A single action potential is generated in response to a pulse of current. Repetitive spiking can be generated only for extremely strong injected currents or not at all.

Response to ramp current input

Picture: Typical response of the three types of neurons to a ramp current input.

Left: Class 1 neuron (can fire at arbitrary frequencies)

Middle: Class 2 neuron (has limited firing frequency range)

Right: Class 3 neuron (does not fire in this case). Note that at the end of the ramp the injected current is larger than the threshold

Ramps, steps, and pulses

Figure: the responses to the three types of inputs.

Ramp: While the current slowly increases, the equilibrium slowly

moves, and the trajectory follows it. When the current reaches I₁, the trajectory is at the new equilibrium, so no response is evoked

because the equilibrium is stable.

Step: When the current is stepped from I₀ to I₁, the system behaves according to dynamics at I₁ with the initial condition set to the

location of the old equilibrium.

Shock: Finally, shocking the neuron results in an instantaneous increase of its membrane potential to a new value. This shifts the initial condition horizontally to a new point marked by the white square (without changing the phase portrait) and results in a spike response.

Ramps, steps, and pulses

Classification based on bifurcations

Figure: Classification of neurons according to the bifurcation

of the resting state.

Summary of neurocomputational properties

Integrators and resonators

Integrator neurons act as low-pass filters, resonator neurons as band- pass filters in response to zap current injection

Left: Zap current injection (Sinusoidal current with increasing frequency) into a resonator neuron.

Right: integrator and resonator impedance profiles.

Integrators and resonators

Picture: short current pulse injections with 5,10,15 ms intervals.

Integrators prefer high frequency inputs, resonators prefer inputs at

Integrators and resonators

Resonators can detect coincidences and resonant inputs, like correctly timed synapses on the soma, or correctly timed synaptic inputs in the dendritic tree. In the latter case the propagation speed across the

dendrite becomes important too, as a distal input can take a few milliseconds to arrive at the soma.

Spiking threshold

Thresholds are always manifolds (curves in a two dimensional system), not single points in the phase space.

An integrator neuron is near a saddle-node bifurcation, so there is a saddle point with its stable manifold (called separatix).

Depending on the prior activity of the neuron and the input, it can generate a subthreshold potential (yellow area), or generate an action potential (white area)

If a system has a subcritical Andronov-Hopf bifurcation with an unstable limit cycle separating the resting and the spiking states, then this limit cycle acts as a threshold manifold (fig. b).

Other systems can produce arbitrary size action potentials, thus we define a threshold set (fig. c) in the phase space which corresponds to manifolds which cause large EPSPs, but not full-size action potentials. Since there is no precise number when a sufficiently big EPSP counts as an action potential, there is no single threshold manifold.

In some models the threshold set is only approx. 0.0001mV wide, which is not relevant for biophysically realistic situations. This is called the quasithreshold phenomenon (fig. d)

Spiking threshold in resonators

Minimal current to elicit spiking

Integrator cells have well-defined minimal currents that elicit spikes: The current input changes the phase portrait, so that the resting state (fig. A, denoted by white rectangle) moves into the spiking area.

Resonator cells do not have this minimal current, since they can produce arbitrary size EPSPs.

Resonators can be excited by inhibition

The negative pulse deactivates a fast low-threshold resonant current, e.g., K⁺ current, which is partially activated at rest. Upon release, the membrane is depolarised, since the K⁺ channels are inactivated.

This is not possible in integrator cells, since inhibitory pulse increases the distance to the threshold manifold (fig. a).

In resonator cells a sufficiently strong inhibitory pulse can push the state of the neuron beyond the threshold set, thereby evoking a rebound

action potential (fig. b).

Post-inhibitory facilitation in resonators

Post-inhibitory facilitation means that a subthreshold excitatory pulse can become suprathreshold if it is preceded by an inhibitory pulse.

In resonators an inhibitory pulse can bring the system closer to the excited state (right figure), thus a previously subthreshold impulse is enough to elicit an action potential. In this case timing plays an important role, producing the interesting effect that well-timed

Summary

In this lesson we learned:

• How to describe the behavior of one-dimensional neuron models: steady- state I-V diagrams, stability analysis, equilibrium points, phase portraits, bifurcation diagrams. With the help of these we were able to determine equilibrium points of the model and predict its behavior.

• Next we analyzed 2-D systems with the help of vector fields. We found that repetitive activity is represented by limit cycles in the phase space.

• We analyzed bifurcations and split them into two major categories: Saddle- node, where the unstable saddle and the stable node approach each other, coalesce, and terminate each other; and Andronov-Hopf bifurcation, where an unstable limit cycle shrinks to an equilibrium and makes it lose stability.

• Next we reviewed Hodgkin’s classification of neurons: Type 1 excitability is when APs can be generated with arbitrary low frequency; Type 2 refers to systems where the frequency of action potentials is band-limited; and Type 3 are systems where a sustained current input can evoke only one action potential.

• We classified neurons into two types according to their response to time- varying input: Integrator and resonator type neurons.

• Finally we analyzed spiking thresholds and found that resonators don't have an explicit threshold, and can be excited by inhibition.

Suggested reading

Books:

• Eugene M. Izhikevich: Dynamical Systems in Neuroscience. MIT Press.

• G. Bard Ermentrout and David H. Terman: Mathematical Foundations of Neuroscience. Springer.

• Steven H. Strogatz: Nonlinear Dynamics and Chaos. Westview Press.

• Perko Lawrence: Differential Equations and Dynamical Systems.

Springer-Verlag, New York.

• Yuri Kuznetsov: Elements of Applied Bifurcation Theory. Springer- Verlag, New York.

Articles:

• Hodgkin A.L. (1948) The local electric changes associated with repetitive action in a non-medullated axon. Journal of Physiology 107:165–181.

• Izhikevich E.M. (2000) Neural Excitability, Spiking, and Bursting.

International Journal of Bifurcation and Chaos. 10:1171-1266.