Before describing the two hippocampal network models of CA3 and CA1 areas that we developed, we first need to examine the firing properties of single cell models. Below we review two computational models that have been developed for pyramidal CA3 and CA1 neurons and one model for fast spiking interneurons. Since two of these, namely the Pinsky-Rinzel and the Wang-Buzs´aki models, will be used as single cell models in our hippocampal networks, we examine their individual behavior and show that they accurately reproduce many of the aforementioned firing properties.
5.3.1 The Traub Model
A complex mathematical model of a guinea pig CA3 pyramidal neuron was introduced by Traub et al. [221] and was used in a series of studies of hippocampal properties [214].
Chapter 5: Network Models of the Hippocampal CA3 and CA1 Areas
The “Traub model”, a modification of an earlier version [210], is a Hodgkin-Huxley- kinetics cable model consisting of 19 compartments: 8 basilar dendritic, 10 apical dendritic and 1 axosomatic. Throughout these compartments, the conductances of 6 ionic currents with compartment-specific densities are distributed: a leakage current, a sodium current, a high-voltage activated calcium current and 4 potassium currents: a delayed rectifier (IK(DR)), an A-type transient (IK(A)), a slow Ca2+-dependent after- hyperpolarisation (IK(AHP )) and a fast Ca2+-dependent current (IK(Ca2+
)). The model kinetics for all the above currents, except for IK(Ca2+
)were obtained from voltage-clamp data from isolated hippocampal pyramidal neurons. Calcium channels are mostly lo-
cated in dendritic compartments while Na+ channels were mostly in perisomatic com-
partments. The intracellular Ca2+ concentration is given by:
d[Ca2+]/dt = −αICa2+− β[Ca2+] (5.3.1)
with α and β fixed parameters.
The model succeeds in reproducing three different types of firing pattern of CA3 neurons, when stimulated by depolarising constant currents or brief current pulses (Figure5.2A- B):
• When the model neuron is depolarised by a low current, it fires in a low-frequency bursting mode. The bursting frequency increases with the current magnitude. • For large enough depolarising current, the firing mode switches to high-frequency
tonic firing.
• In between these modes, a mixture of bursts and tonic spikes exists.
Therefore, the model manages to reproduce the two characteristic CA3 firing patterns mentioned in Section 5.2. A third firing mode, tonic dendritic Ca2+ spiking, was also observed when the cell was stimulated by a large depolarising current applied in the middle or distal dendrites. Namely, the large depolarisation caused tonic Na+ spikes in the soma that initiated respective Ca2+ currents in the dendrites.
By increasing the maximal conductance of the IK(DR) current, decreasing those of the dendritic ICa2+ and IK(C) currents and adjusting the conductances distributions, the
CA3 neuron model could be transformed into a model of a CA1 pyramidal cell. After these alterations, somatic depolarisation leads to tonic firing with adaptation, whereas dendritic depolarisation leads to bursting.
Chapter 5: Network Models of the Hippocampal CA3 and CA1 Areas
A
B
C
Figure 5.2: A: The three firing patterns of a CA3 pyramidal cell as reproduced by the Traub model. B: Frequency-current curve for the bursting (left) and tonic spiking (right) firing modes of the Traub model. The insets show representative dendritic (top) and somatic (bottom) voltage traces of the 2 modes. Taken from [221]. C: Comparison of frequency-current curves for the Traub and the Pinsky-Rinzel models. Taken from [165].
An extended version of the CA3 Traub model was later developed [212], containing 64 somatic and dendritic compartments with branching dendrites, an axon initial segment and 4 axonal compartments. The kinetics and the distributions of the ionic currents remained similar as before and so did the single cell firing patterns.
5.3.2 The Pinsky-Rinzel Model
The same year that the extended version of the Traub model was published, a much simplified version of this CA3 pyramidal cell model was developed by Pinsky and Rinzel
[165]. Their goal was to reduce the original Traub model while retaining its main
characteristics and properties. Their approach was to keep the same currents and gating kinetics but reduce the number of compartments.
The Pinsky-Rinzel model is a two compartment Hodgkin-Huxley cell with one dendritic compartment where all the calcium and calcium-dependent currents are located, and one axosomatic compartment with the sodium and the delayed rectifier potassium cur- rents. The connection between the two compartments is regulated by a simple coupling
Chapter 5: Network Models of the Hippocampal CA3 and CA1 Areas
conductance whose value was shown to be critical in achieving the same behavior as the Traub model. Again, the intracellular calcium concentration is given by an ODE of the same form as in eq. (5.3.1). Altogether, this model consists of only 8 variables, compared to 120 variables for the original Traub model.
The basic firing modes of the hippocampal cell were successfully reproduced by this model as well:
• The model exhibits periodic bursting with a low somatic current or with a stronger constant dendritic input. The burst duration is primarily determined by the time course of dendritic Ca2+ build up while the interburst interval depends on the time courses of the hyperpolarising currents IK(Ca2+) and IK(AHP ). The burst is
initiated by a sodium spike in the soma which triggers a slower dendritic calcium spike and the burst is formed by a “ping-pong type” current flow between the somatic and dendritic compartments [51].
• Greater depolarisation leads to periodic somatic spiking without active dendritic spikes. For specific coupling conductance and compartment sizes the resulting firing frequency as a function of the applied current (f-I curve) resembles that of Traub’s (Figure 5.2C). In between the bursting and tonic firing modes, aperiodic behavior and spike doublets are observed, as in Traub’s model.
• Even higher current injection eventually leads to annihilation of spikes and a steady depolarised state is reached.
• Dendritic input results in bursting with frequency increasing along with the current amplitude.
5.3.3 A Modification of the Pinsky-Rinzel Model
In order to simulate the firing properties of CA1 pyramidal cells, we introduce a slight modification in the Pinsky-Rinzel model’s parameter values. We aim to replace the typical CA3 intrinsic bursting, under a somatic-injected depolarising current, with tonic firing exhibiting frequency accommodation [8,134]. Since the somatic bursts are super- imposed on a longer dendritic calcium spike, we keep all parameter values the same as
in the original model with the only exception being the Ca2+ maximum conductance
Chapter 5: Network Models of the Hippocampal CA3 and CA1 Areas
We simulated the original Pinsky-Rinzel model in “Brian” and, after accurately repro- ducing its firing properties (not shown), we implemented the gCa2+parameter alteration.
Depolarising the somatic compartment with low currents now results in tonic firing with a frequency that quickly attenuates to a constant value. We calculated the firing rate of the cell through the first, second and last interspike interval after a 2 sec run of the model. As shown in Figure 5.3, the firing rate increases linearly with increasing current and the accommodation becomes faster. Greater depolarisations result in an initial burst that is followed by accommodating tonic firing. The slope of the f-I curve, calculated through the last interspike interval is approximately 30.35 Hz/nA which is in good agreement with the 34.1 Hz/nA slope estimated through single cell recordings [126].
Figure 5.3: f-I curve for the Pinsky-Rinzel model of the CA1 pyramidal cell, after the parameter alteration described in Section5.3.3. Frequencies are calculated by the inverse of the first (triangles), second (crosses) and the last interspike interval (squares) after a 2 sec simulation. The slope of the least squares fit on the last interspike intervals is 30.35 Hz/nA. The insets show the membrane potential of the cell when depolarised in the dendrite by constant current of 0.2 nA (upper) and 0.8 nA (lower).
5.3.4 The Wang-Buzs´aki Model
The Wang-Buzs´aki model is a simple single-compartment Hodgkin-Huxley cell with
only the basic spike-generating currents (leakage, sodium and delayed rectifier potas- sium currents) [228]. It is well suited for modelling fast spiking interneurons, such as hippocampal basket cells, since it can fire repetitive spikes at high frequencies and each spike is followed by a very brief refractory period. Its firing rate increases very steeply as
Chapter 5: Network Models of the Hippocampal CA3 and CA1 Areas
a function of the applied current and can reach high frequencies of several hundred Hz even with depolarising pulses of a few µA/cm2 (see Figure4.9in the previous chapter).