Input design and validation

When identifying a system that does not belong to the model structure, the input used to perform parameter estimation heavily influences the quality of the identified model;

the parameter estimates ˆθN are only optimal with respect to the deterministic input ik

used to excite the system.

Even for simple linear systems, it is well known that a model that is estimated and successfully validated on one class of signals may not be well validated on another class of signals (for a simple analytical example, see [77, Example 8.2]). A good type of training input is thus one that results in a model that generalizes well towards (that is validated sucessfully on) different classes of input.

Inputs used in the literature

In the context of biophysically realistic neuronal models, [33] studied how well models trained on either steps, ramps, or constant-mean white noise generalized towards input types not used for training. It was found that models trained on combined data

5.3 Identification procedure 127

from multiple experiments with ramp and step inputs generalize well towards noisy input; on the other hand, models trained solely on noisy input do not generalize well towards ramp and step inputs. The conclusion seems to be that using noisy input in the identification of neuronal systems might not be the best choice for input design, and indeed many works in the field of parameter extraction for biophysically realistic neuronal models limit the type of the excitation to either steps or ramps.

However, there is evidence in the field of system identification that filtered white noise (and random phase multisines) are good signals to identify nonlinear oscillators [26]. Since periodic oscillatory behavior is a possible regime of excitable behavior, this evidence should at least in part apply to those systems as well. Furthermore, filtered white noise has an advantage as an input signal over ramps and steps in that it allows arbitrarily long data-gathering experiments to be conducted, whereas when using ramps and steps, multiple experiments of a limited length need to be conducted.

It is worth mentioning that some authors have used swept-sine signals (also known as chirp or zap), given by sinusoids with increasing frequencies in time, to characterize the frequency response of single neurons [118, 110]. One issue with this type of signals, as shown in [26], is the fact that they may not provide a sufficiently rich excitation for nonlinear systems possessing limit-cycles.

Input design for identifying excitability

One class of inputs that does not seem to have been explored in depth in the literature is the combination of user-defined white (or filtered) noise with a deterministic ramp (note that the user-defined input noise in ik should not be confused with the unmeasured membrane current noise ek). We have previously described this type of input in Equation (5.20), in the context of bifurcation detection. We reproduce it here for convenience:

ik= imin+imax− imin

Kts

kts+ ηk, 0 ≤ k ≤ K,

We argue that the class of inputs above ideal to identify excitability, particularly when using a very general model structure such as the one introduced earlier. The key lies in a judicious selection of the value of the variance of the noise ηk.

Keeping the variance of ηk sufficiently low is useful to ensure that the linearized behavior of the system in the subthreshold voltage range v < v^∗ is well captured by the estimated model (in Figure (5.6), for instance, this is the range where v < −60 mV).

By estimating well the linearized behavior of the ground truth system, the estimated model will not have spurious bifurcations of the stable equilibrium that defines the

subthreshold regime of the system. Furthermore, estimating well the linearized behavior guarantees that the right bifurcation is learned by the estimated model.

In the nonlinear identification stage, the variance of ηk should, however, not be too small. First, it is important to keep the signal-to-noise ratio as large as possible.

Equally important, the noise should be large enough to allow identification of the limit cycle attractor underlying spiking behavior. To identify such attractors, it is important that the the state-space of the original system be explored as much as possible close to the limit cycle [26], and for that to happen, the input noise level should be sufficiently high.

Periodizing the ramp

In order to design effective input signals to identify excitable behaviors, we need to be able to capture the hysteresis of the F-I curve that defines Type II and Type II^∗ excitability. For that reason, it is important that we not only apply an increasing ramp, but also a decreasing ramp that starts from a high current.

We thus propose an input signal in which white or filtered white noise is added to a deterministic sawtooth wave — a periodic combination of increasing and decreasing ramp signals. This input is given by

i_k = to be such that K/2 is an integer, and P is the number of periods of the wave. The signal ηk is white or coloured noise.

In addition to allowing for the identification of excitability, a periodized ramp allows for arbitrarily long experiments (in which case the convergence results presented before can be applied).

Validation

Despite the fact that our identification strategy focuses on identifying the internal system (5.2) with an output error nonlinear model structure, the ultimate objective of the identification procedure is to be able to successfully simulate the excitable behavior of the closed-loop system (5.1)-(5.2), and not just that of the internal system. This means that, for validating the identified model, we need to compare the output of