Figure 18: Some of the copying results (schematic) of an associative case (narrow sense). Left: what the subject was shown. Middle: the way the subject copied it. Right: theoretically, alternate possibility.

are spatially very distinct.

In summary, we regard these findings as an indication that perception - or recognition evolvement and representation - may be more individual to an object and does not follow a standard evolvement scheme.

3.7 Neuronal Level

Until now we have only addressed the system level. But what may we learn from the neuronal level? We start with the electrical properties of the nerve cell membrane.

Subthreshold dynamics Below the spiking threshold, the membrane behaves roughly like a resistor-capacitor (RC) circuit, in which the capacitance, Cm, represents the membrane capacitance; the resistor, Rleak, expresses the ‘leakiness’ of the membrane; and the voltage node, Vm, represents the membrane voltage (figure 19a). If one stimulates the nerve cell membrane with a step current of low amplitude (see two ‘I’ traces), its membrane voltage will gradually increase and then saturate. After offset of the stimulating current, the membrane voltage will bounce back (see corresponding two ‘Vm’ traces). How fast this increase and decay occurs depends on the parameter values for the resistance and membrane capacitance. A neuron’s membrane capacitance can be reliably estimated, but its membrane resistance is not alway easy to determine, especially in case of the small neocortical neurons. The decay times range from a few milliseconds to a couple of tens of milliseconds (Johnston and Wu, 1995). If the decay time is

rather long, a neuron may function simply as an integrator, because it would continuously add up synaptic input. In contrast, if the decay time is short, then the neuron may function as a coincidence detector: synaptic input is only added up, when it occurs almost simultaneously, otherwise synaptic input leaks away before other synaptic input adds up. In which one of these two modi a cortical neuron operates, is still an on-going debate (e.g.(Koenig et al., 1996; Koch, 1999)).

Synaptics The synaptic responses can be very diverse. There exist the simple excitatory and inhibitory synapses, that cause a transient increase or decrease in the membrane potential respectively. A schematic excitatory postsynaptic potential (EPSP) is shown in figure 19b on the left side. Its duration lasts several milliseconds. Two excitatory synapses sum up linearly as shown in figure 19b (upper right). There exist also voltage-dependent synapses that produce non-linear summation: they generate a larger synaptic response, when the membrane potential has already been elevated by other stimulation (figure 19b, lower right). This type of response can be interpreted as multiplicative (gray trace indicates mere summation). There are a number of other synaptic interactions, as for example divisive inhibition. This variety of synaptic responses could therefore perform quite a spectrum of arithmetic operations (Koch, 1999).

Dendritics The tree-like dendrite of a neuron collects synaptic signals in some way. In a first instance, the dendrite may be regarded as a passive cable conveying the synaptic signals that were placed on it. The propagation properties of such a dendritic cable have been characterized to a great extent by Rall (Rall, 1964). For practical simulation purposes, a dendritic tree is discretized into so-called compartments, each of which simulates a patch of membrane by a RC circuit. The compartments are connected by a horizontal (or axial) resistance, Raxial, which simulates the internal, cellular resistance. Figure 19c (left) shows two compartments connected by a horizontal resistance. If synaptic input is placed for example in the left compartment (voltage node Vm1), then it will spread towards its neighboring compartments and decay in amplitude due to the horizontal resistance and the membrane resistance of each RC circuit. The schematic in figure 19c (right side) shows this propagation decay for a synaptic input placed on Vm1. Given these decay properties, a dendrite may be regarded as a low-pass filter. But there maybe more complex processing going on in a dendrite than just mere synaptic response integration along a passive cable. Because many dendrites contain also voltage-dependent processes that could cause quite a number of different operations. For instance, a dendritic tree sprinkled with the voltage-dependent ‘multiplicative’ synapse just mentioned previously, could carry out sophisticated computations (Mel, 1993). Or a den-

Figure 19: Schematic neuronal dynamics. a. Resistor-capacitor dynamics represent the nerve cell membrane dynamics (below spiking threshold). Left: RC circuit: Cm: membrane capacitance; Vm membrane potential: Rleak: membrane resistance. Right: The response of Vm to two (superimposed) step currents. b. Synaptic response and integration. Left: typical excitatory postsynaptic potential. Right: linear summation (above), multiplicative summation (below). c. Compartmental modeling of a piece of dendritic cable. Left: A cable is well simulated by discretizing it into separate RC circuits connected by axial (‘horizontal’) resistors, Raxial; Right: Synaptic response decay along compartments.

dritic tree containing voltage-dependent channels like the ones in the soma (see next paragraph) could actively propagate a synaptic input toward the soma. Furthermore, spike waves have been discovered that run up and down the dendrite (e.g (Svoboda et al., 1997). Thus, there could be a substantial amount of active processing taking place

erated. Once the membrane voltage crosses a certain threshold, an action potential is generated that is the result of a marvelous interaction of subtly timed ionic currents (Hodgkin and Huxley, 1952). Such an impulse is generated within 1-2 milliseconds. If the neuron is stimulated continuously, for example with a step current of sufficient magnitude, then the neuron spikes repetitively. There is a large number of models that mimic this spiking behavior ranging from detailed models, that simulate the exact dynamics of the ionic current, over to oscillatory models and to simplified spiking models. From this wide range of suggested neuronal functions and models we here mention only one type, the integrate-and-fire (I&F) model (Koch, 1999). It is the simplest type of spiking neuron that incorporates explicit subthreshold and spiking dynamics.

Figure 20: Integrate-and-fire (I&F) variants. a. Perfect I&F neuron: C represents the membrane capacitance, V the membrane potential, the box with the pulse icon represents the spiking unit which activates a switch (S) resetting the membrane potential. b. Leaky I&F neuron: it contains a resistance (R) in addition. Modified from Koch, 1999.

The most elementary I&F variant is the perfect I&F neuron (figure 20a). It merely consists of a capacitance, representing the membrane capacitance, and a spike-generating unit possessing a thresh-