Iterated function systems in the hippocampal CA1

Abstract

How does the information of spatiotemporal sequence stemming from the hippocampal CA3 area affect the postsynaptic membrane potentials of the hippocampal CA1 neurons? In a recent study, we observed hierarchical clusters of the distribution of membrane potentials of CA1 neurons, arranged according to the history of input sequences (Fukushima et al Cogn Neurodyn 1(4):305–316, 2007). In the present paper, we deal with the dynamical mechanism generating such a hierarchical distribution. The recording data were investigated using return map analysis. We also deal with a collective behavior at population level, using a reconstructed multi-cell recording data set. At both individual cell and population levels, a return map of the response sequence of CA1 pyramidal cells was well approximated by a set of contractive affine transformations, where the transformations represent self-organized rules by which the input pattern sequences are encoded. These findings provide direct evidence that the information of temporal sequences generated in CA3 can be self-similarly represented in the membrane potentials of CA1 pyramidal cells.

Appendices

Appendix 1

Brain slices of six Wister rats (4–5 weeks old) were made according to the standard procedure reported by Tsukada et al. (2005). The brain was sliced at an angle of 30–45° to the long axis of the hippocampus, with a thickness of 300μm. Before recording, slices were kept in bath solutions (142 NaCl/2 MgSO₄/2.6 NaH₂PO₄/2 CaCl₂/26 NaHCO₃/10 glucose (mM), bubbled with 95% O₂/5% CO₂) at room temperature for at least 60 min.

The slice was then placed in the recording chamber. Neurons were visualized with an IR-DIC camera (C2741079H, Hamamatsu, Japan). Recording electrodes were pulled from borosilicate glass and the resistance was 5–8 MΩ. Recordings were done at 28–30°C. The internal solution of the recording electrode was prepared following the method of Magee (2001), and contained 120 KMeSO₄/20 KCl/10 HEPES/10 EGTA/4 Mg₂ATP/0.3 TrisGTP/14 Tris₂phosphocreatine/4 NaCl (mM, pH7.25 with KOH). Whole cell patch-clamp recording was obtained from the soma of pyramidal cells in CA1 using an electrical amplifier (CEZ-3100, Nihonkoden, Japan). Signals were filtered at 10 kHz, sampled at 25 kHz, and stored (micro 1401, CED, England). The starting voltage of the recorded neurons was between −57 and −44 mV, and the membrane potential was maintained at a voltage between −67 and −76 mV by current injection to the soma before electrical stimulations using two theta glass electrodes (TST-150-6, WPI, Florida).

Appendix 2

When an order of two or more groups with respect to the value of a variable is specified by an alternative hypothesis, the Jonckheere-Terpstra (JT) test (Terpstra 1952; Jonckheere 1954) can be employed instead of the Kruskal-Wallis test (Sheskin 2004). In this study, we used the JT test to evaluate whether conditional distributions of responses of an individual cell to input pattern sequences with length greater than one were ordered in a self-similar manner according to the similarity of the input pattern sequences. Specifically, for each input pattern sequence with (k − 1) length i ₁···i _k−1, the JT test was applied to a set of four conditional distributions $\{G_{i}= [i_1\cdots i_{k-1} i]; i={\bf 1},\ldots, {\bf 4}\}$ sharing the common recent history i ₁···i _k−1. The alternative hypothesis is that the {G _i } have the same order as the four conditional distributions $\{[i]; i={\bf 1},\ldots, {\bf 4}\}$ to the four input patterns. More precisely, if the order of medians m _i of [i] is m ₁ < ··· <  m ₄ and the median of the distribution G _i is θ _i , the alternative hypothesis is θ₁ ≤ ··· ≤ θ₄ with at least one strict inequality. After the JT test, one-sided comparisons between θ _i <  θ _j (i < j) were conducted using the Bonferroni-Dunn (BD) multiple comparison test (Dunn 1964; Sheskin 2004). For all tests in this study, the significance levels were set at p = 0.05. In the BD multiple comparison test, this corresponds to the assignment of statistical significance at p-values 0.05/6 (₄ C ₂ = 6) to each one-sided comparison.

Appendix 3

To know the mechanism for the emergence of contractive affine transformations, we describe a continuous time evolution of somatic membrane potentials with spatiotemporal inputs. For simplicity, we treated the case in which the membrane potentials stayed below the firing threshold. Then, we conducted data-fitting of its time course during each input interval using the leaky-integrator neuron model,

$$ \tau_m(dV/dt) = -V + V_{syn}, $$

(3)

where V [mV] is the somatic membrane potential expressing deviation from the resting potential, τ _m [ ms = kΩ μF] is the decay time constant of the somatic membrane potential due to leakage, and V _syn [mV] is the effect of synaptic input on the somatic membrane potential. The solution V(Δt) of Eq. 3 in elapsed time Δt from an input time is expressed by the following formula:

$$ V(\Updelta t) = V(0)\exp(-\Updelta t/\tau_m) + A(\Updelta t), $$

(4)

where V(0) is the somatic membrane potential at input time, and A(Δt) is the convolution of V _syn (s) and exp(−s/τ _m ) from s = 0 to Δt, that is, A(Δt) = τ ⁻¹ _m ∫ ^Δt₀ V _syn (Δt −s) exp(−s/τ _m ) ds. The time course of the effect of synaptic input V _syn (Δt) was simply given by an α-function (Gerstner and Kistler 2002):

$$ V_{syn}(\Updelta t) = q_r \tau_s^{-2}(\Updelta t -\delta) \exp\left(-\tau_s^{-1}(\Updelta t-\delta)\right) \Uptheta(\Updelta t-\delta) $$

(5)

where q _r [mV ms = kΩ μC] is the total change in somatic membrane potential due to the total effective charge that is injected in the neuron via its synapses, τ _s [ms] is the rise and decay time constant of the α-function, δ[ms] is the transmission delay and Θ(s) is the Heaviside step function with Θ(s) = 1 for s > 0 and Θ(s) = 0 otherwise. We assumed that the term V _syn represents a collective input-triggered effect on the somatic membrane potential, which includes not only excitatory synaptic inputs but also feedback and feedforward inhibitory inputs.

For each data set of responses {V _Δt (n) ; 0 ≤ Δt ≤ 30} (S(n) = 2, 3 and 4) of cells in the sub-threshold group, we estimated four parameters (τ _m , τ _s , q _r , δ) in Eqs. 4 and 5, using nonlinear regression with the least-squares criterion. However, recording data at an early phase in each input interval, [0, t ₀), were omitted for estimation to avoid contamination due to electrical stimulus artifact. The least-squares minimization was performed using nls package with a port algorithm of version 2.5.1 of the R software (R Development Core Team 2007), which iteratively searches for a solution in a given bounded range from a given initial condition using information of the functional derivative.

A solution for the four parameters was searched for from the following intervals: 0.1 <  τ _m <  500, 0.1 <  τ _s <  min(30, τ _m ), 0.1 <  q _r  < 500, 0.1 < δ < min(10, t ₀) and t ₀ ≥ 6. These search intervals were taken so widely in order that the solution not saturate so much. For each data set, twenty initial conditions were prepared. We accepted a set of convergent values of the parameters as the solution if its root mean square error was minimum among all sets of convergent values and was less than 0.15. In almost all data sets, however, the convergent values of the parameters were not sensitive to the choice of the initial conditions.