^{1}

This paper provides an overview of different types of models for studying activity of nerve cells and their networks with a special emphasis on neural oscillations. One part describes the neuronal models based on the Hodgkin and Huxley formalism first described in the 1950s. It is discussed how further simplifications of this formalism enable mathematical analysis of the process of neural excitability. The focus of the paper’s second component is on network activity. Understanding network function is one of the important frontiers remaining in neuroscience. At present, experimental techniques can only provide global recordings or samples of the activity of the huge networks that form the nervous system. Models in neuroscience can therefore play a critical role by providing a framework for integration of necessarily incomplete datasets, thereby providing insight into the mechanisms of neural function. Network models can either explicitly contain individual network nodes that model the neurons, or they can be based on representations of compound population activity. The latter approach was pioneered by Wilson and Cowan in the 1970s. Finally I provide an overview and discuss how network models are employed in the study of neuronal network pathology such as epilepsy.

A fundamental and famous model in neuroscience was described by Hodgkin and Huxley in 1952 [

An additional problem with these detailed models is that, because of their complexity, they cannot be analyzed mathematically and may not therefore lead to deeper understanding of the process it models. Further model reduction is required before mathematical analysis can be employed. For example, the Hodgkin and Huxley model is a four dimensional nonlinear model (the dynamic variables are membrane potential

A further simplification for simulation of nerve cell activity is the leaky integrate-and-fire (IF) model [

When modeling neuronal networks, one can start from the network nodes, that is, the neurons, or make direct assumptions at a higher organization level, that is, the population activity. In the first case, the bottom-up approach, one must decide how to model the nodes. Simple on/off switches are the most abstract models of neurons in networks. The underlying hypothesis of this approach is that, for some of the aggregate behavior, the network properties themselves are of principal interest while the details of the cellular properties (beyond having active and inactive states) are irrelevant [

A different family of network models describes aggregate behavior of neuronal populations. Although the components of these models are derived from or inspired by the neuronal function, they do not explicitly consist of a network of individual agents. This approach was pioneered by Wilson and Cowan [

Understanding the function of neuronal networks is a frontier in current neuroscience. Our lack of fundamental understanding of network function in the nervous system prevents us from linking neuronal activity to higher level physiologic and pathologic behavior. Although compound activity of huge networks can be captured by the EEG or possibly by functional magnetic resonance imaging (fMRI), current experimental techniques in neuroscience cannot record the individual activity of cells forming larger networks (>1000 neurons) with sufficient resolution in the time domain (<1 ms). Due to the lack of such techniques, there is a real need for theoretical and computational models in neuroscience because they can provide a framework for the integration of experimental data. In addition, models provide an efficient tool for generating and testing series of hypotheses; to cite Abbott [

The purpose of this paper is to provide a summary of different models for neural activity with a focus on oscillatory patterns, to discuss the relationship between modeling approaches, and to place them in a logical and a historical context. Section

The Hodgkin and Huxley (H&H) equations [

(a) Diagram of a biomembrane that separates the inside and outside of the nerve cell. See text for further explanation. (b) Membrane equivalent circuit with

Because ions are charged, an ion pump creates a potential difference and can therefore be modeled as a potential source (

In the H&H formalism, the potential difference between the fiber’s in- and outside, the membrane potential (

A linearized version of the H&H equations is useful to examine subthreshold oscillatory properties of the biomembrane around a resting or holding potential (say

In order to linearize the nonlinear equations, we rewrite (

For the gating parameters ((

Now we consider the potassium current and model small perturbations and show we can model this by using a two branch electrical circuit with resistors

Diagrams of equivalent circuits representing ion channels. (a) A two branch equivalent circuit for a linearized potassium channel. The potassium channel has properties similar to an inductor. A change in current flowing through an inductor creates a time-varying magnetic field inside the coil; this induces a potential that opposes the change in current that created it. In the potassium channel, a change of K^{+} current is opposed by the change in membrane potential and the associated change in K^{+}-conductance caused by the current. (b) Equivalent circuit for linearized sodium activation and inactivation. (c) A part of the linearized membrane model for the sodium channel with negative values for both

First we restate, from (

We can now use (

Now we can directly relate the linearized potassium channel to the circuit in Figure

Combining the above we find that

Now we apply the linearization to inward current

From the three terms in (

Combining the above we find

So we can successfully model small perturbations in the sodium channel with the equivalent circuit in Figure

In the previous sections, it was shown that the channels in a linearized version of the H&H equations can be represented by equivalent circuits consisting of a network of resistors and inductors (Figure

Electrical equivalent circuit of the linearized H&H equations.

The inductor channel in the circuit represents the stabilizing effects of the outward current activation (e.g., potassium) and inward current inactivation (e.g., sodium). The resistor and capacitor represent membrane resistance and capacitance plus corrections for negative impedances. Since we look at small perturbations around an equilibrium, which we conveniently set at zero, the driving forces of the difference between Nernst potential and resting potential can be ignored.

The pair of differential equations that govern the circuit in Figure

In electrophysiology, cellular properties are often probed with hyperpolarizing and depolarizing current steps. The unit impulse response, the derivative of the unit step, is the inverse Fourier transform or inverse Laplace transform of the system’s frequency response or transfer function, respectively. Since we deal with a linearized system with current

Bode plot of the linearized model depicted in Figure

An interesting observation is that the Bode plot in Figure

The H&H representation and its linearized version can be seen as a basis for many different neuronal models. Of course one could either add complexity to the H&H formalism or further reduce it. The following sections summarize representative examples of both types.

Using the H&H formalism one might extend the sodium and potassium channels of the model neurons to also include different types of potassium channels (such as the A-current, a transient potassium current), persistent sodium channels, and different types of Ca^{++} channels. In addition to the voltage sensitive channels, it is common practice to also include Ca^{++} and/or Na^{+} sensitive channels. In addition to the addition of channels, the cell model can also be divided into many multiple coupled compartments (e.g., [

One can simplify the H&H equations while preserving the nonlinearity, or one might use the linearization results described in Section

Overview of the dimensionality of cellular models.

Amplifying | Resonant | |||||
---|---|---|---|---|---|---|

Membrane potential |
Inward current activation | Inward current inactivation | Outward current activation |
Leak current capacitance |
Physiological AP mechanism | |

Hodgkin and Huxley [ |
1 | ( |
( |
( |
Yes | Yes |

FitzHugh-Nagumo [ |
1 | Instantaneous |
Generic recovery | ( |
Yes | Yes |

Morris and Lecar [ |
1 | Instantaneous |
No | ( |
Yes | Yes |

IF [ |
1 | No | No | No | Yes | Generator |

RF [ |
1 | No | No | Simplified ( |
Yes | Generator |

QIF [ |
1 | Generic amplification |
No | No | Yes | Reset |

Izhikevich SMC [ |
1 | Generic amplification |
No | Simplified ( |
Yes | Reset |

AP: action potential; L: inductor; IF: integrate and fire; QIF: quadratic integrate and fire; RF: resonate and Fire; SMC: simple model of choice;

Example of an electronic circuit of an IF neuron model using analog components. The membrane is modeled by the R and C components (orange), the threshold is implemented by the OP-Amp as a comparator (the threshold is the value at the potentiometer, orange). The reset function is performed by the Reed relay. The diodes and 10 k

(1) ^{+} current. In the original equations (

(2)

Historically, the RC-based integrate-and-fire model was introduced long before the H&H formalism in 1907 by Lapicque [

Phase space representation for the QIF neuron. The relationship of membrane current, proportional to the derivative of

Finally, one can combine the QIF scenario with the RCL-circuit to produce a model that is capable of both resonance and spike upstroke generation. This is defined as the simple model of choice (SMC) by Izhikevich and was shown to be able to produce a plethora of neuronal behavior (e.g., [

We can consider the different models and summarize their relationships by counting the dimensions (Table

Network models containing concrete nodes come in a wide variety, ranging from nodes represented by an on/off switch to a detailed multicompartmental cell model with multiple ion channels. Similarly, the connections between the nodes can be modeled by relatively simple rules or they can be simulated by biophysically realistic synaptic ion channels. In general, the purpose of these models is to provide simulation results that can be compared to a result from mathematical analysis and/or an experimental measurement.

The first simple network model using coupled switches was described by McCulloch and Pitts [

At each time step, the state of

The update of

This update process is a stochastic variant of one of the rules described by McCulloch and Pitts [

A model similar to the Ising spin lattice was described by Hopfield [

As described in Section

More complicated cell models have also been employed to investigate network activity. Recently, Izhikevich and Edelman [

Using the H&H formalism combined with approximation of the cell morphology by segments, one can create realistic representations of neurons. Addition of ligand-sensitive, synaptic channels allows one to create networks. Equation (

The detailed model of neocortex outlined in Figure

Model of Neocortex. The detailed model (after [

Not only the oscillatory activity pattern itself but also the sudden transition between nonoscillatory and oscillatory network behavior can be understood from bifurcation analysis of a global model of two coupled excitatory-inhibitory populations as depicted in Figure

The network models discussed in this section do not explicitly simulate each network node, but start directly by modeling populations of nerve cells. Some of the models explicitly use the mean field approach commonly employed in statistical mechanics, others create neural populations with functionality inspired by the single neuron function. The final result of these approaches can be fairly similar since they both usually contain excitatory and inhibitory components (the Ising spin model is an exception). As I will point out repeatedly, this results in a strong similarity between the equations that result from the various network models and even between these and the equations describing single neurons.

Two- or three-dimensional models of the Ising-spin lattice, where the spins represent the neurons, are not easy to solve. A mean field approach is the simplest approximation to describe such a system. In the lattice, each node experiences the sum of the magnetic forces created by the other nodes and, if present, an external magnetic field. In the mean field approach, we replace the internal field generated by the spins by its average value. It is beyond the scope of this paper to derive the mean field relationship, which can be found in many physics textbooks (e.g., [

The transition between bi- and mono-stability occurs at the bifurcation where the straight line

Properties of the mean field equation of the Ising model. (a) The stability diagram. (b) Bifurcation diagram. (c) The cusp catastrophe shows the dependence of magnetization

Stability diagram

Bifurcation diagram at

Cusp catastrophe

The 1st model by Wilson and Cowan [

Diagram of the Wilson-Cowan Model containing excitatory and inhibitory populations. The coupling strength between the populations is denoted with constants

In Wilson and Cowan [

Phase plane representations of the Wilson-Cowan model during equilibrium, oscillatory, and limit-cycle activities. These plots correspond to the parameters in Figures 4, 10, and 11 of the original 1972 paper [

To study resonance phenomena, we will consider the effect on

The model described by Lopes da Silva et al. [

(a) Diagram of the Model (after [

In a similar fashion we include the effect of the inhibition in the diagram in Figure

We can substitute ^{−1}, ^{−1}, ^{−1}, ^{−1},

The Wilson-Cowan Equations [

Jansen’s neural mass model [

Neural mass models. (a) A diagram of the model of a cortical unit employed by Grimbert and Faugeras [

The models created by Freeman in the 1980s used a combined network modeling and experimental approach to study the electrical activity in rabbit olfactory system. Although Freeman’s approach may not have been directly inspired by the Wilson-Cowan equations and/or the models by Lopes da Silva and co-workers, the fundamental building blocks of his model are neural masses. The appeal of Freeman’s approach is that it is strongly based on the anatomy of the olfactory system, and that he relates modeling results to electrophysiological experiments. As compared to the previous models, a unique addition is that it contains latencies L1–L4, modeling the conduction delays occurring in between populations [

In his models, Freeman describes different population levels. Within the most basic population type, which is defined as K0, he includes pulse-to-wave conversion, linear spatiotemporal integration, wave-to-pulse conversion, and conduction of the output. This basic population is modeled with a 2nd order linear ODE coupled to a nonlinear static sigmoid function, the Wiener cascade (Figure

(1) K0: a subset of noninteracting neurons, either all excitatory (

(2) KI: coupled K0 networks create the KI level. In Freeman’s terminology, two

(3) KII: this type of population arises when a

(4) KIII: when KII sets are interconnected, a KIII set is created. Note that there is just one such a set in Figure

Simulations of Freeman’s model were implemented in Matlab and typical results are depicted in the bottom panel in Figure

The electrical signal that can be recorded from the scalp, the electroencephalogram (EEG), reflects the currents generated by the brain’s nerve cells. If we ignore electrical signals from other more remote sources (e.g., muscle, heart) and nonbiological artifacts, we can assume that the EEG is the weighted sum of the underlying neural network in which the weights are determined by the geometry. The EEG signal is used both in research and clinical settings. Because a single EEG signal includes the activity of millions of nerve cells, the relationship between smaller networks of the brain and the EEG signal is not necessarily a trivial one. Therefore this relationship has been the target of many modeling studies; the framework developed by physicist Paul Nunez is an example. Because it is generally thought that the slower synaptic potentials, especially those of the neocortical pyramidal cells, contribute most to the EEG signals on the surface of cortex and scalp, Nunez’ model focused on describing synaptic activity across cortex (e.g., [

The model presented by Nunez is based on interaction between cortical volume units. The output

The external input

An excellent summary of macroscopic modeling is provided in Jirsa and Haken [

The excitatory (

Diagrams of the model described by Jirsa and Haken [

(1)

(2)

The inclusion of a conduction delay is correct when assuming active propagation of synaptic input, for passive propagation (at the speed of light) such a delay is not needed. For the remainder of the derivation by Jirsa and Haken [

In the model, cortical connectivity is described by exponential distributions of the form

(1)

Using the simplifications indicated by the horizontal curly brackets above we have the expression for the excitatory synaptic activity:

The next step is to write the result for

Now we complete this step and write the expression in the form of a convolution with respect to space and time

Now we take advantage of the convolution form of (

Simulation of (

With the exception of the spin model and Hopfield’s approach (Section

Recently, Benayoun et al. [

The next step is to consider the so-called master equation for this process. In contrast to the deterministic approach above, we now use a stochastic approach and describe the probability that there are

Under this assumption, the probabilities describing the dynamics around state

when the next reaction (any of the

what kind of reaction (

In order to resolve these questions, Gillespie follows the system of reactions from any time

Gillespie’s approach for Monte Carlo simulation of the master equation. (a) The joint PDF that determines the reactions

Because the master equation can be used to recover the deterministic rate equation, the stochastic approach is more general than the mean field models we discussed in Section

We now plug in (

A recent, novel approach to solve the master equation for a neuronal network model used the equation of motion of the generator function in operator notation

Now we get

Epilepsy is a serious neurological disorder characterized by spontaneous recurrent seizures. In the electroencephalogram (EEG) of patients with epilepsy, one may observe seizures (ictal events) and interictal events: for example, epileptic spikes or spike-waves. The real problem is that about 30% of the current population of 60 million patients with epilepsy do not respond to any treatment [

Although there are many clinical types of seizure [

normal and epileptiform states,

prototypical oscillations and bursts, and

transitions between the states.

Can seizure onset be modeled with a Hopf bifurcation? (a) A partial complex seizure spreads and grows in amplitude, suggesting one might model it with a supercritical Hopf bifurcation. (b) A generalized seizure occurs suddenly and at full amplitude, similar to a subcritical Hopf bifurcation. (c) Note that the same seizure as in (b) can also be explained by a perturbation in a bistable regime associated with the subcritical Hopf bifurcation. In this case the transition into the seizure state is not a bifurcation but due to a change of basin of attraction (e.g., [

From these properties, a few general conclusions about the nature of the model can be determined. First, a model of network activity must be at least two dimensional to explain oscillations. Second, transitions between normal and epileptiform activity can be modeled by a bifurcation in a nonlinear dynamical system. Of course, there are many bifurcations that could explain sudden onset and offset of seizure-like oscillations. Examples using the simplest candidate: co-dimension-1 bifurcations transitioning a system between steady state and oscillatory behavior, the sub- and supercritical Hopf bifurcations [

Some of the models employed in epilepsy research go beyond the minimalistic approach of two dimensions. The detailed simulation models with compartmental neuron models with realistic ion channels may have hundreds of parameters. Independent of the level of complexity of the network models, it is important to know if the neuron model itself is capable of oscillatory activity so that the individual network node can be a pacemaker for oscillation.

Here we examine the population models we described in Section

Bifurcations, caused by a gradually varying excitatory coupling (blue, top trace), creating a seizure-like epoch in population activity governed by the Wilson and Cowan Equations (Excitatory population–red and inhibitory population–green).

The modeling approach by Lopes da Silva et al. [

Linearized models do not support bifurcations and therefore fall short in explaining a critical aspect of epilepsy (seizure onset and offset). The linearized model of Nunez [

Diagram of string models used for EEG. (a) A string attached at its ends but unattached to local structures; the associated detail shows the forces acting on a small piece

Finally we examine the result (

In the previous sections, I presented a variety of modeling approaches used in neuroscience. As shown in Section

In general, the interest in and acceptance of modeling in neuroscience is growing (e.g., [

Because of the necessary simplifications required in modeling, there are limitations in the models reviewed here; they neglect many aspects that play important roles. A critical part of the dynamics of neuronal networks is determined by synaptic plasticity, that is, the change in network connection strength. These changes play a crucial role in the process of learning but, when pathologic, may also contribute to diseases such as epilepsy. It was more than a half century ago that Hebb [

Finally, one important conclusion is that there is not “a best approach” when modeling neural function. For example, nonlinear models can help us to understand specifics such as sudden transitions, but linear ones are more suitable to analyze subthreshold properties such as oscillation. The linearized versions of the models also make it easier to see similarities between the models across different levels of organization. Another example: complex models can be studied with simulations, they are more complete since they can deal with more parameters than the more abstract models used for mathematical analysis. However, both have a place and they can be complementary. Analysis of simplified mathematical models can generate fundamental insight in spite of the fact that they lack detail. Simulation can be too complex to directly generate such fundamental insight into the interaction of the ongoing processes, but they have the advantage to be closer to reality. This property appeals to the experimenter since it may produce data with a temporal-spatial resolution that is not (yet) feasible experimentally. In this context, the ultimate goal in understanding neural networks in physiological and pathological states, is to create a framework of experimental models, detailed simulations, and mathematical formalisms that allow us to understand and to predict dynamics of network activities including state transitions; that is, results in one model can be used to inspire work in another type of model. In the case of analyzing a heterogeneous network disease such as epilepsy, modeling can provide an overview of hypothetical network mechanisms involved in ictal activity that can be employed as a guide for experimental and clinical investigation.

In order to linearize the nonlinear equations, we rewrite (

For example in Richardson et al. [

The following Matlab scripts were used to create some of the Figures in this paper. They can be copied in the Matlab command window, or (via the Matlab editor) saved as an m file.

% Bodeplt.m

C=1e-10;

R=2e8;

RL=2e7;

L=2e6;

gam=(1/R)+(RL*C/L);

eps=(1/L)*(1+(RL/R));

B=[1 RL/L];

A=[C gam eps];

w=0 : 1000;

freqs(B,A,w).

% IsingModel.m

% h - external magnetic field

% J and n are coupling strength and number of

clear;

close all;

J=1;

n=1;

%%%%%%%%%%%%%%%%%% Stability Diagram

% To find this diagram we can solve for

% 0=T*atanh(m) - (J*n*m+h) and

% d/dm(T*atanh(m))=d/dm(J*n*m+h)

figure; hold;

i=1;

for mm=-1 : .01 : 1

end;

plot(h_star, Tc, ‘k’)

title(‘Stability Diagram’)

xlabel(‘h - Magnetic Field’)

ylabel(‘T - Temperature’)

%%%%%%%%%%%%%%%% 3D

figure; hold;

for mmm=-1 : .01 : 1;

h=T*atanh(mmm) - J*n*mmm;

plot3(h,T, mmm)

end;

plot(h_star,Tc, ‘k’)

axis([-1.5 1.5 -.1 1.5 -1.1 1.1])

title(‘3D diagram - blue; Stability Diagram – black’)

xlabel(‘h’)

ylabel(‘T’)

zlabel(‘m’)

view(-20, 70).

% ModelSpectrum_LdS1974.m

% Fig. 8 in Lopes da Silva et al., 1974

clear

% Parameters (see p. 36 in Lopes da Silva et al., 1974)

A=1.65e-3; % in V

B=32e-3; % in V

C1=32;

C2=3;

a1=55;

a2=605;

b1=27.5;

b2=55;

qe1qi1=4.55e6;

P=1; % assume random input (i.e. same for all

K=A*B*C1*C2*qe1qi1*(a2-a1)*(b2-b1);

n=0;

for f=0 : .1 : 30 % Frequency range

end;

figure;

semilogy(F, abs(V))

xlabel(‘Frequency (Hz)’)

ylabel(‘abs(V(jw))’)

title(‘Fig. 8 Lopes da Silva et al., 1974’).

% netsim.m

clear;

close all

% Parameters and initial values

N_cells=10

N=round((rand(1)/2+.1)*rand(1,N_cells)); % randomized initial state of vector N

AD(1)=sum(N); % initial value for the deterministic case

alpha=10; % decay rate a -> q (set around 1/100 ms)

f_tilde=5; % q -> a (set around 5 Hz)

dt=.001; % time step for the deterministic model

T=1; % Epoch (in s)

timeD=0 : dt : T; % determinsitic timebase

steps=floor(T/dt); % determinsitic # of steps

% Deterministic model

% - - - - - - - - - - - - - - - - - - -

% this simulation based on kinetic rate equation

% note that equilibrium is at dA=0 - -> A_equilibrium

for i=1 : steps

end;

% Equivalent Stochastic model

% - - - - - - - - - - - - - - - - - - - - - - - - - - -

% this simulation based on stochastic model

% Specific parameters and initial values

% background of the method can be found in

count=1; % counter for the AS array

cum_t=0; % initial time

timeS(count)=cum_t; % stochastic time base

while cum_t < T % Main LOOP

if P(i)==0; P(i)=f_tilde; end; % all inactive cells become f

i=i+1;

if (F(i)>=r(2)); mu=i; pick=1; end;

end;

% plot results

figure; hold

plot(timeD, AD, ‘k’)

plot(timeS(1 : length(AS)), AS)

title (‘Deterministic model (black); stochastic (blue)’)

xlabel(‘time (s)’)

ylabel(‘# of Active Cells’).

% HopfBifurcation.m

% Depict a Hopf Bifurcation in the WC Eqs.

% x - E pop; y - I pop

% P - External input to E pop

clear;

close all;

x(1)=0;

y(1)=0;

P=4.0;

WE(1)=1;

dWE=.01

dt=.1;

T=300;

N=T/dt;

tim=0 : dt : T-dt;

for i=1 : N-1

end;

figure;

subplot(2, 1, 2), plot(tim, x, ‘r’)

hold;

subplot(2, 1, 2), plot(tim, y, ‘g’)

axis([0 300 -.05 0.55])

subplot(2, 1, 1), plot(tim, WE)

axis

title(‘Hopf Bifurcation in the WC Equations: E-red, I-green, External Input to E-blue’)

xlabel(‘Time’).

This work was supported by the Dr. Ralph and Marian Falk Medical Research Trust. Thanks are due to Albert Wildeman for providing the data for Figure