3.1 Measuring SIE in layer 5 pyramidal cells
Using somatic whole-cell recordings from layer 5 pyramidal cells, we injected a fluctuating current to mimic background synaptic activity, causing the neuron to spike at approximately 10 spikes/s. On top of the fluctuating current we added a sequence of EPSC-shaped currents (sEPSC) with a fixed time course (see Sect. 2, Fig. 1) and varied their peak amplitude (the range of spike output rates in these experiments was 8–20 spikes/s). Figure 1 illustrates the injection protocol and the resulting voltage response of the neuron. We then used the sequence of times of the input sEPSC as an input “spike train” and computed the mutual information between this spike train and the output spike train emitted by the neuron to obtain the SIE for every current amplitude. The SIE as a function of current amplitude is depicted in Fig. 1b for four different neurons. For a small input (e.g. 0–400 pA) the SIE is typically very small (less than 2 bits/s). Such an input is illustrated in the top example trace (Fig. 1a: 300 pA). In this case the input sEPSC is smaller than the standard deviation of the background current and cannot be identified by inspection. Moreover, for a given output action potential it is sometimes difficult to tell whether it was caused by the sEPSC or not. Nevertheless even such a small current sometimes clearly influences spike generation (see second and fifth inputs in the example) and changes the probability of the neuron to emit an action potential. This effect, though too small to be noticeable by eye, is detected by the SIE. For larger input amplitudes the SIE grows linearly with input size. When the current is large enough to rise above the noise an output spike will frequently follow the input sEPSC. For the example shown in Fig. 1a (1,600 pA), 85% of sEPSCs are followed by an output spike within 4ms. Typically a failure to trigger a spike occurs because the input just followed an output spike while the neuron was refractory, or the input followed a random large hyperpolarization (see third and fifth inputs in the example). For even larger sEPSCs (>2nA) the SIE saturates at approximately 35–40 bits/s (for the bin size of 3 ms we use here).This saturation occurs because the information is bounded by the minimum of the entropies of the input and the output. Intuitively one cannot gain more information than there is in the output or than is supplied by the input (see also London et al. 2002, Fig. 2). The reason that only one cell shows this saturation is that we did not drive the neurons with input amplitudes large enough to show this effect (this is generally detrimental to the recording and the health of the neuron).
The relationship between SIE and input size is therefore nonlinear, and follows a sigmoidal trajectory. This relationship qualitatively agrees very well with earlier predictions based on simulations (London et al. 2002) where its properties are discussed in more detail. This non-linear shape is in contrast to the linear relationship expected between cross-correlation and input size as described in Herrmann and Gerstner (2001) and is primarily due to the nonlinear properties of the mutual information measure. In the following sections we would like to explore how well a simplified model can capture the relationship between SIE and input amplitude described above.
3.2 Constructing simplified integrate and fire models to match experimental data
Many successful sophisticated methods for constructing simplified models of spiking neurons are described in this issue, and in the literature (Brunel and Latham 2003; Fourcaud-Trocme et al. 2003; Rauch et al. 2003; Keren 2005; Jolivet et al. 2006; Badel et al. 2008). Nevertheless we deliberately choose here to take the most simplistic view and fit the recordings with a leaky integrate-and-fire model in order to demonstrate the power of the SIE approach. The model has three passive parameters (R membrane resistance in Ω, τ membrane time constant in s and V
rmp resting membrane potential in V).
The model has three additional parameters to describe spikes: θ, voltage threshold in V; V
AHP, the voltage to which the membrane potential is reset after each spike; and τ
ref, the refractory period during which the membrane potential is clamped to V
AHP. Figure 2 describes the fitting process (see Sect. 2). We split the process of fitting the model into two stages. As the current-voltage relationship for these cells is linear over a wide range of subthreshold potentials (Fig. 2a), we choose to use a standard minimization procedure to fit the subthreshold response in the first stage. In the second step we fit the threshold, V
AHP and refractory period based on the spike triggered average curve (see also Sect. 2).
We used one trace with a small sEPSC amplitude (<300 pA) for training. We then used this model on the rest of the input currents (with different amplitudes). Predicting every spike with such a simple LIF model is impossible. The gamma coincidence factor (GCF) (Kistler et al. 1997) measures how many spikes of the model coincide with the real spikes of the neuron (with the required tight condition of 2 ms accuracy), normalized to chance level (i.e. if the two were spiking in their mean rate but completely independent of each other). For the input currents used in this study the model achieved a GCF of 0.5 ± 0.1. This score is expected for the standard linear integrate-and-fire model but is lower than that of adaptive integrate-and-fire models presented in this Special Issue which achieve a GCF of ∼0.8.
3.3 Comparing the effect of synaptic input in the experiment and in the model
As the SIE is measuring the efficacy of the synaptic input using only the input and output spike trains, clearly, as pointed out above, if the model could predict the output of the cell perfectly for any given input, then the SIE for the model and the real neuron would be identical. However, as we see in Fig. 2 for the integrate and fire model some of the original spikes are missed, some are spurious and some are shifted compared to the real cell.
Figure 3a depicts the SIE as a function of sEPSC current amplitude for one of the neurons presented in Fig. 1b. The model shown in Fig. 2 was used to integrate the same input currents, which were used in the experiments. The SIE computed for the model shows a remarkable similarity to the SIE computed for the data. Similar results were obtained for two of the other cells presented in Fig. 1 (RMS—2.3, 3.1, 3.13 bits/s for cells 1, 2 and 3, respectively). The explanation for this close agreement is quite interesting. Both the fluctuating input current and the input spike trains are random and thus the neuron and the model have statistically similar output spike trains (even though they do not overlap on the millisecond time scale). This makes their entropies similar. The LIF model, while poor in predicting the spike caused by the background fluctuating current (e.g. GCF = 0.5 when the sEPSC is very small; see Fig. 2) is pretty good in predicting spikes that are driven by the input sEPSC. This makes the conditional entropy of the model and the neuron quite similar as well. All in all the SIE which is the difference between the entropy and the conditional entropy corresponds closely between the model and the experiments. In that sense the SIE is less sensitive to the precise timing of the spikes arising from the “background” current, and more sensitive to the timing of spike in relation to the input sEPSC. For cell 4 the agreement between the SIE of the model and of the experiments was not as good as in the other cells shown in Fig. 3b (RMS: 6.7 bits/s). Interestingly, this cell was firing bursts (typically of two spikes) in response to current injection at the soma, in contrast to the other three neurons. An example trace of recording from this neuron is shown in the inset, together with the corresponding model trace. Clearly it is impossible to describe this behavior with such a simple model. With our fitting strategy the model has the disadvantage that V
AHP is relatively high (−43mV) and τ
ref is very short (2 ms) in order to account for the bursting activity. However, in contrast to the real neuron the model has no memory, and thus produces many more spikes per burst than the real neuron. Moreover, the real neuron tended to be quiet after a burst, while the model does not. The result shows that the model often reacts to the input with more spikes than the real cell, which increases the entropy of the output. However when the input is known (to the estimation algorithm), it accounts for these spikes, which reduces the conditional entropy. The increase in entropy and reduction in conditional entropy compared to the real data, results in an overestimate of the mutual information, hence the overestimate of the SIE when computed with the model.
3.4 Using mutual information to evaluate the predictive power of the model
Till now we computed the mutual information between the input and the output of both the experimental data and of a model to evaluate how well the model captures the input- output relationship of the real cell. We can also use the mutual information in a different way: to evaluate directly how well the model predicts the output. The traditional way this is done (especially in this Special Issue) is using the GCF (Kistler et al. 1997). Similarly, we can measure the mutual information between the output spike train of the model and that of the real neuron. If the two are completely independent we should obtain zero mutual information. If, on the other hand, the model output is precisely that of the neuron, then the mutual information will be the entropy of that output spike train. Normalizing the mutual information by the entropy of the neuron’s output will yield a measure between 0 and 1, similar to the GCF.
It is important to understand that there are differences between the two measures. On one hand, if one is interested in a model that predicts every spike precisely, then of course the GCF provides a good measure to compare models. On the other hand, if the model predicts the exact output spike train but with a time shift of, say, 3 ms, then the GCF will be very small even though the output is by almost any measure identical. Similarly, if the model predicts a pair of output spikes for every single output spike of the neuron, again, the two spike trains would carry almost exactly the same amount of information, but the GCF will give a rather poor score for the performance of the model (because only ∼50% of the spikes are predicted). The mutual information is, however, agnostic about such transformations, and thus suitable as a complementary method to evaluate the quality of models.
Figure 4 shows the GCF as well as the normalized mutual information for the two cells presented in Fig. 3a, b. For the first cell the two measures have very similar shape, only the mutual information gives lower scores for the model compared to the GCF. Very similar matches were obtained for the other two regularly spiking cells. For the bursting cell, however, the two methods gave different results. The GCF is relatively independent of the input (i.e. sEPSC) amplitude. The mutual information, on the other hand, grows linearly with the input size. Curiously, this is the model that performed least well in capturing the input-output relationship of the neuron by overestimating the SIE. This result hints that indeed the mutual information can pick up the relationship between the prediction of the model and the real data that the GCF ignores. However, in this case there is also the complication that the model that performs worst, yields better mutual information. There is however quite a simple explanation for this. In Fig. 3b we estimated the mutual information between the input and the output. As we showed, the model overestimated this information because for large inputs it tends to “over burst” compared to the real neuron (i.e. fire more spikes in a burst per input). Thus, when we know the input, we can reduce the entropy of the models output even more than we can do for the real neuron. Here when we compute the mutual information between the model’s output and the neuron’s output, the input is not known, but, for large enough sEPSCs, we already know that the model almost certainly produces a burst per each input. So to some extent the model output acts in a similar way as the input given to the synapse. In other words, bursts of the real neuron are well predicted by bursts of the model, even though there are more spikes in each burst, and this increases the mutual information between the model’s output and the neuron’s output.