The Izhikevich equations represent a simple model of spiking neurons as a two- dimensional system having a fast voltage variable and a slower recovery variable (Izhikevich, 2003; Izhikevich, 2007; Izhikevich and Edelman, 2008). The membrane recovery variable accounts for the activation of K+ ionic currents and the inactivation of
slower variable has a sigmoid-shaped nullcline. The principal equations of the dimensional form of the phenomenological model are:
C dv/dt = k ( v – vr ) ( v – vt ) – u + I
du/dt = a { b ( v – vr ) – u }
if v ≥ vpeak then v ← c , u ← u + d
where v is the membrane potential, u is the membrane recovery current variable, t is time, I is the input current, C is the membrane capacitance, k is the rheobase parameter, vr is
the resting membrane potential, vt is the instantaneous threshold potential, vpeak is the
soma’s spike cutoff, a is the recovery time constant (the decay rate), b is the input resistance parameter (the sensitivity of the recovery variable to subthreshold fluctuations of the membrane potential), c is the soma’s after spike voltage reset and d is the after- spike reset of the recovery variable (the outward minus inward currents activated during the spike and affecting the after-spike behavior).
The alternative form:
may also be used. With full synaptic kinetics, the current is computed as:
I (t) = − Idendritic− Isynaptic
with the synaptic current given by:
Isynaptic = gAMPA ( v – 0 )
+ gNMDA { [ ( v + 80 ) / 60 ]2 / ( 1 + [ ( v + 80 ) / 60 ]2 ) } ( v – 0 )
+ gGABAA ( v + 70 )
+ gGABAB ( v + 90 )
+ Igap
and the conductance given by:
g (t) = g0 e– t / τ
with
g0 = g (t) + c
With various choices of parameters a, b, c and d, the model can exhibit all known types of firing patterns. Our parameter value choices are provided in Table 6.3. Some of our implementations have included the glutamatergic excitatory regular spiking (RS) cell, the most typical neuron in cortex, as well as the GABAergic inhibitory fast spiking (FS) interneuron cell. We typically exclude short-term synaptic plasticity, NMDA and GABAB contributions and dopamine-modulated dendritic spike-timing-dependent
plasticity (STDP), but include the excitatory AMPA and inhibitory GABAA presynaptic
currents.
However, we have found that even the simpler, alternative form yields excellent results. This can best be described as a pulse-coupled neural network (PCNN), with the firing of a presynaptic neuron instantaneously changing the variable v by a predetermined synaptic connection weight.
We create a population of Izhikevich cell models to explore synchronization dynamics. We have explored several topologies and again, even the simplest are effective. For the results that we will present here, we implement ten V4 cells, each receiving a noisy injection current. These cells all have excitatory projections to an IT cell. The IT cell has an excitatory connection to an interneuron, presumably at the level of IT, which may or may not have inhibitory feedback connections to the V4 cells. We again use genetic algorithms (Goldberg, 1989) and searches of parameter space to find connection weights
parameter value C RS: 100 FS: 20 PCNN: x µF/cm2 k RS: 3 FS: 1 PCNN: x vr RS: –60 FS: –55 mV vt RS: –50 FS: –40 mV vpeak (soma) RS: 50 FS: 25 mV v2 PCNN: 0.04 mV v1 PCNN: 5 mV v0 PCNN: 140 mV a RS: 0.01 FS: 0.15 PCNN: 0.02 (excitatory) 0.1 (inhibitory) b RS: 5 FS: 8 PCNN: 0.2 c (soma) RS: –60 FS: –55 PCNN: –65 d RS: 400 FS: 200 PCNN: 8 (excitatory) 2 (inhibitory) τ 5 (AMPA) 6 (GABAA) c 10 (AMPA) 4 (GABAA)
All simulations were carried out in a Microsoft Windows XP Professional SP2 environment on an Intel® Pentium® 4 CPU running at 2.80 GHz with 3.00 GB of RAM. All models were constructed using the MATLAB application development environment (version 7.9.0.529 R2009b) and the associated Curve Fitting Toolbox (version 2.1), Genetic Algorithm and Direct Search Toolbox (version 2.4.2), Image Processing Toolbox (version 6.4), Neural Network Toolbox (version 6.0.3), Optimization Toolbox (version 4.3), Signal Processing Toolbox (version 6.12), Statistics Toolbox (version 7.2) and Wavelet Toolbox (version 4.4.1).
6.3 Results
The Morris-Lecar model, the FitzHugh-Nagumo model within a winnerless competition network and the Izhikevich model within a feedback network have been implemented.
6.3.1 The Morris-Lecar Model and Recognition
The validation of our Morris-Lecar model cell is shown in Figure 6.1. The top left plot depicts this cell’s response to a constant current application. The bottom left plot shows the frequency components of this same response. We see that the injected current
Figure 6.1 – Morris-Lecar model validation. The top left plot depicts a Morris-Lecar cell’s response to a constant current application. The bottom left plot shows the frequency components of this same response. The plot on the right provides the oscillation frequency of the cell in response to a wide range of input currents – typical behavior of a Type I oscillator.
currents. This behavior – increased stimulus intensity (current injection) resulting in increased frequency of response over a wide range – is typical behavior of a Type I oscillator and is necessary for our type of intensity-driven recognition.
In Figure 6.2 we show a Morris-Lecar cell’s response to selected images. This Morris- Lecar cell models an IT cell, which integrates specific information about the 2- dimensional boundary shapes of multiple contour fragments (the V4-like cell inputs, responding to an image’s iso-curvature segments). Its parameters, including Gaussian constituents and coefficients, are selected to match those of an IT-like cell created in Chapter 5. The optimal set of synaptic conductances for this cell is found using a genetic algorithm. The sample images on the left side of the figure are presented to the Morris- Lecar cell at time t = 0, producing the responses on the right side. This cell, tuned to respond to guitars, prefers guitar images above images in other categories.
Note that this situation is not entirely biologically realistic. The stimulus is presented at time t = 0 and the Morris-Lecar IT-like cell appears to respond instantaneously (i.e., without regard for synaptic transmission delays through retina, primary visual cortex, extrastriate cortex, etc.). Also, the cell’s spiking is persistent – in contrast to the diminishing response seen in cortex. However, the IT-like cell’s differential firing rate – the primary objective of this exercise – is accurate and reflects what the Connor group has observed in IT (Brincat and Connor, 2004; Brincat and Connor, 2006).
Figure 6.2 – Morris-Lecar cell’s response to selected images. This Morris-Lecar cell models an IT cell, with inputs from V4 cells. Sample images are on the left side. Image numbers are given. The corresponding Morris-Lecar cell responses (with image presentation at time t = 0) are on the right side. Response frequency is shown in red. This cell, tuned to respond to guitars, prefers guitar images above images in other categories.
Ten example Kanwisher natural images from each of the seven sampled categories (axes, cats, fish, guitars, handsaws, hats, and scissors) are selected. In Figure 6.3, we show the histogram of responses of one similarly-created Morris-Lecar IT cell to each of these images. Again, this cell is tuned to respond preferentially to guitars. This is similar to our work in Chapter 5 with “mathematical” cells and is reminiscent of the study of a single unit in the left posterior hippocampus / medial temporal lobe of epilepsy patients with depth electrodes by Cristof Koch (Quiroga et al., 2005). Here, the cells were activated exclusively by different views of Jennifer Aniston, for example, and not Julia Roberts, etc. Note that these cells are not designed to be, nor do they behave like, “grandmother” cells. They simply respond well to cells within a single category, at the exclusion of others. We have created many Morris-Lecar IT-like cells similar to these that respond preferentially to each of the seven categories in our sample space.
6.3.2 The FitzHugh-Nagumo Model, Spatiotemporal Patterns and
Winnerless Competition
Our winnerless competition network topography with nine FitzHugh-Nagumo model neurons is shown in Figure 6.4. Notice the unsymmetrical connectivity of the cells and their expectations of curvature values from the images’ bounding contours at forty-degree increments.
Figure 6.3 – Morris-Lecar cell’s response to each image. The 70 sample images are grouped into 7 categories (with 10 in each category). All 7 categories (axes, cats, fish, guitars, handsaws, hats, and scissors) are represented. The height of each bar represents the response of the Morris-Lecar cell (modeling an IT cell) to each of these images. Again, this cell was tuned to respond preferentially to guitars.