Discussion of the model

We used three aspects of the model to describe experimental data. (i) The linear resonat-ing model is well suited to cover subthreshold responses. (ii) In combination with a static renewal threshold, the differences between the firing statistics of resonant and nonresonant neurons could be quantitatively captured. Especially the multimodality of ISI distributions in resonant cells versus the monomodality of these distributions in nonresonant cells were reproduced by the model. (iii) The modified model, with a stochastic nonrenewal threshold, proved to capture quantitatively the ISI distributions as well as the negative ISI correlations in resonant and nonresonant cells. The spike probability in the nonrenewal model takes the spike history into account, such that a recent occurrence of spikes lowers the firing probability.

We observed that the intensity of intrinsic noise always increased towards the threshold (Fig.7.2). This effect is most likely due to the stochasticity in the opening and closing of ion channels (for a review on channel noise see White et al., 2000). The influence of noise on

7.7 Discussion 135

spike initiation has been shown to be important (Schneidman et al., 1998), we therefore chose to make the spike probability a dynamical variable. The increased level of noise around the threshold as well as the influence of spike history on spike initiation are therefore incorporated into a single step.

Overall, the nonrenewal model is simple and captures the essential properties of the subthreshold frequency preference and interspike correlations. It may prove useful for network simulations, where subthreshold properties of individual neurons and correlations in their spike trains may be of importance. Other nonrenewal models that capture correlations and adaptive effects have been suggested before. These models modify, either dynamically or stochastically, the absolute threshold value (Chacron et al., 2004) or explicitly model currents that prolong the interspike interval in dependence on the spike history (Benda and Herz, 2003;

Izhikevich, 2004). We have no reason to assume, that these nonrenewal models cannot also represent our experimental data.

The results of this chapter provide new insight how intrinsic cell properties acting on the subthreshold level of signal processing affect the spike generation in the entorhinal cortex.

Whereas clustering of action potentials clearly is a hallmark of resonant neurons, additional spike-induced dynamics further modifies the firing statistics. Simple threshold models, that combine the subthreshold dynamics with nonrenewal spike initiation, are easy and efficient to use in network simulations and faithfully reflect the intrinsic neuronal frequency selectivity.

8

Concluding remarks

In this thesis various aspects of the non-Markovian first passage time problem have been investigated, using a combination of analytical, numerical and experimental methods. Two main issues have been addressed in this text:

I. the elaboration of analytical methods to treat the non-Markovian first passage time problem,

II. the exploration of mechanisms leading to complex spike patterns in neurons.

I. Non-Markovian first passage time problem

Recent experiments (Diau et al., 1998; Desmaisons et al., 1999; Nowakowski and Kawczyński, 2006) revealed essentially non-Markovian character of the escape dynamics in many physical systems. This explains a growing theoretical interest in the challenging non-Markovian first passage time problem. The distinctive features of the non-Markovian escape dynamics are the time dependent escape rates and the sensitivity to initial conditions. Complex multimodal structures are characteristic for the distribution of the first passage time in the non-Markovian case. Therefore, habitual Kramers approach reveals its inherent limitations when confronted with non-Markovian situations. Several analytical approaches to the non-Markovian first passage time problem have been proposed in the literature. These include the method of optimal fluctuation (Soskin et al., 2001) and approximations based on the generalized renewal equation (Wilemski and Fixman, 1974; Likthman and Marques, 2006).

We hope, this work provides a new insight into the non-Markovian first passage time problem. Our analytical approach is based on the theory of level-crossings first developed by Rice (1945). An exact expression for the first passage time density of a differentiable random process can be derived in form of an infinite series of integrals over the joint densities of level-crossings. We demonstrate, how effective and accurate analytical approximations for this series can be obtained on the basis of either truncations or the correlations decoupling. In the former case, a few initial terms evaluated exactly are used to approximate the series (direct

truncations and Padé approximants). In the latter case infinitely many terms evaluated approximately lead to a closed analytical expression for the first passage time density (Hertz and Stratonovich approximations). Our approximations are applicable to processes with differentiable trajectories, but there are no further restrictions on the dimension and form of the dynamical system. We compare the quality of different approximations and ascertain their regions of validity. The approximations complement one another: in regimes of strong noise and low threshold the truncations are more suitable, whereas for a weak noise or high threshold the decoupling approximations are more effective. Overall, the regions of validity for our approximations cover all types of escape dynamics, ranging from almost Markovian to strongly non-Markovian cases.

In the general Markovian case, a single parameter – the constant escape rate – completely describes the statistics of the escape times. In contrast, in the non-Markovian case multi-ple time scales are involved in the escape problem. These time scales are pertinent to the deterministic subthreshold dynamics and are reflected in the statistics of the escape times, for example in the multipeak first passage time densities. Our analytical approach to the non-Markovian first passage time problem allows a closer inspection of these complex multi-peak first passage time densities: the number of pronounced multi-peaks, their position and height are reproduced by the analytical approximations and can be related to the deterministic subthreshold time scales.

The level-crossing approach is rather general. It relates the statistical properties of a freely evolving random process (in the absence of boundaries) to the statistical properties of a point process formed by the times, when this random process crosses a prescribed level.

This general approach is applicable whenever the joint level-crossing rates are finite. Besides the differentiable random processes the level-crossing approach can also be applied to discrete random walks with arbitrary waiting-time distributions. The generalization of this approach to processes with nondifferentiable trajectories may be a further challenging problem. Finally, inspired by negative correlations found in the dynamics of resonant and nonresonant neurons, a very appealing problem is to develop the modified level-crossing approach for a nonrenewal escape dynamics, with the level-crossing rates being dependent on the history of escape times.

II. Spike patterns in resonant and nonresonant neurons

The neural code (Rieke et al., 1997; Koch, 1999) is the central problem discussed in the computational neuroscience. Current theories hypothesize that information is encoded by the spike frequency, the spike times and the correlations between spike times. Insight into the neural code could be therefore gained by understanding the spike patterns generated in neurons.

In this work we investigate the joint influence of deterministic and stochastic properties of neuron dynamics on the spiking behavior. In particular, we study the random spike patterns in resonant and nonresonant neurons. The former exhibit subthreshold resonance with max-imal response amplitude achieved at a finite resonance frequency. The subthreshold response amplitudes in the latter decay monotonically with the frequency. We show in phenomeno-logical neuron models (FitzHugh-Nagumo model and resonate-and-fire model) as well as in

8. Concluding remarks 139

the experimental data obtained from stellate (resonant) and pyramidal (nonresonant) cells in the entorhinal cortex in rat, that the differences in the deterministic subthreshold prop-erties yield qualitatively different spike patterns. Resonant neurons exhibit spike clustering reflected in the multimodal interspike interval (ISI) densities, whereas nonresonant neurons show monomodal ISI densities.

The idea that the subthreshold resonance properties of neurons may shape their spike patterns is intuitive and have been proposed in several studies (Desmaisons et al., 1999;

Pedroarena et al., 1999; Izhikevich, 2001). In this work we systematically investigate this connection and demonstrate, that the subthreshold frequency selectivity of resonant neurons is sufficient to explain the spike clustering. We use the resonate-and-fire model with exper-imentally estimated subthreshold parameters and show, that this simple phenomenological model can quantitatively capture the ISI distributions measured in stellate and pyramidal cells. The ISI density obtained analytically for the resonate-and-fire model in the Stratonovich approximation agrees well with experimental data. The advantage of using an analytical ap-proximation is that it allows to relate the characteristic time scales observed in spike trains (inter- and intra-cluster intervals) to the characteristic time scales of the subthreshold volt-age dynamics (period and relaxation time of the subthreshold oscillations). Moreover, the mechanism shaping the spike patterns in resonant neurons can be easily understood from the simple resonate-and-fire model. This mechanism is expected to be the same in more complex models.

The subthreshold properties alone are not sufficient to completely explain the observed spike patterns: Negative spike-induced ISI correlations, which are found in stellate as well as in pyramidal cells, cannot be explained by the subthreshold properties. To also account for negative correlations, we introduce a novel nonrenewal threshold mechanism in the resonate-and-fire model. Modified model can capture both: the differences in spike patterns as well as similar correlations in resonant and nonresonant neurons. The predictions of the model are in quantitative agreement with experimental data.

Spike clustering and negative correlations might be essential for the neural computation.

Spike clusters may enhance the reliability of synaptic transmission and allow for frequency specific information routing (Lisman, 1997; Izhikevich et al., 2003). Negative ISI correlations were shown to improve the signal detection and information transfer by neurons (Chacron et al., 2001). Understanding of the interplay between subthreshold resonance properties and spike-induced correlations and their joint influence on the information processing by a neuron are therefore very interesting open problems. The precise mechanism leading to the ISI correlations in stellate and pyramidal cells also remains to be elucidated.

A

Experimental methods

A.1 Experimental data

All details of experimental procedures and data acquisition¹ have been previously described in (Erchova et al., 2004; Schreiber et al., 2004). Briefly, the sharp electrode intracellular recordings were obtained from entorhinal cortex neurons in adult Wistar rats (above 300 g, older than three months). We used medial horizontal slices of the retro-hippocampal region (400 µm, at 35^◦C). Synaptic transmission was blocked. The following drugs were added to the aCSF (in µM): 306-cyano-7-nitroquinoxaline-2,3-dione (CNQX), 60 DL-2-amino-5-phosphonopentanoic acid (APV), 5 bicucculine and 1 CGP (all from Sigma-Aldrich, Deisen-hofen, Germany). All recordings were done in current-clamp mode using sharp micropipettes filled with 2 M potassium acetate containing 1% biocytin (75–100 MΩ). The experiment and acquisition were controlled by custom software written in LabView via a digital-analog converter card (AI-16E-4, National Instruments Inc., TX, USA). Data were sampled at 8 kHz and stored for offline analysis. The cells were identified based on their location and elec-trophysiological properties, 25% of recorded cells were histologically identified, as described in Erchova et al. (2004). Only cells that had an input resistance of more than 20 MΩ, a resting potential less then -60 mV, and a spike amplitude larger than 50 mV were used for the analysis.

We used hyper- and depolarizing current steps with a duration of 500 ms, 10 s, 20 s and 30 s to characterize cells and to obtain distributions of interspike intervals. The oscillatory stimuli (ZAP) of 30 s duration were applied to characterize the resonance properties of a cell:

I(t) =I₀+I₁sin[2πf(t)t], (A.1)

with f(t) = fmt/2T. Here I1 = 50 pA is the stimulus amplitude, I0 the constant holding current,T = 30 s the stimulus length, andf_m = 20 Hz the maximum stimulus frequency.

1 All experiments were performed by Irina Erchova (University of Edinburgh).