Theory of single file diffusion in a force field

The dynamics of hard-core interacting Brownian particles in an external potential field is studied in one dimension. Using the Jepsen line we find a very general and simple formula relating the motion of the tagged center particle, with the classical, time dependent single particle reflection R and transmission T coefficients. Our formula describes rich physical behaviors both in equilibrium and the approach to equilibrium of this many body problem.

CAMKII activation is not required for maintenance of learning-induced enhancement of neuronal excitability

Pyramidal neurons in the piriform cortex from olfactory-discrimination trained rats show enhanced intrinsic neuronal excitability that lasts for several days after learning. Such enhanced intrinsic excitability is mediated by long-term reduction in the post-burst after-hyperpolarization (AHP) which is generated by repetitive spike firing. AHP reduction is due to decreased conductance of a calcium-dependent potassium current, the sI_AHP. We have previously shown that learning-induced AHP reduction is maintained by persistent protein kinase C (PKC) and extracellular regulated kinase (ERK) activation. However, the molecular machinery underlying this long-lasting modulation of intrinsic excitability is yet to be fully described. Here we examine whether the CaMKII, which is known to be crucial in learning, memory and synaptic plasticity processes, is instrumental for the maintenance of learning-induced AHP reduction. KN93, that selectively blocks CaMKII autophosphorylation at Thr286, reduced the AHP in neurons from trained and control rat to the same extent. Consequently, the differences in AHP amplitude and neuronal adaptation between neurons from trained rats and controls remained. Accordingly, the level of activated CaMKII was similar in pirifrom cortex samples taken form trained and control rats. Our data show that although CaMKII modulates the amplitude of AHP of pyramidal neurons in the piriform cortex, its activation is not required for maintaining learning-induced enhancement of neuronal excitability.

Probing dynamics of single molecules: Nonlinear spectroscopy approach

A two level model of a single molecule undergoing spectral diffusion dynamics and interacting with a sequence of two short laser pulses is investigated. Analytical solution for the probability of n=0,1,2 photon emission events for the telegraph and Gaussian processes is obtained. We examine under what circumstances the photon statistics emerging from such pump-probe setup provides new information on the stochastic process parameters and what are the measurement limitations of this technique. The impulsive and selective limits, the semiclassical approximation, and the fast modulation limit exhibit general behaviors of this new type of spectroscopy. We show that in the fast modulation limit, where one has to use impulsive pulses in order to obtain meaningful results, the information on the photon statistics is contained in the molecule's dipole correlation function, equivalently to continuous wave experiments. In contrast, the photon statistics obtained within the selective limit depends on the both spectral shifts and rates and exhibits oscillations, which are not found in the corresponding line shape.

Persistent ERK activation maintains learning-induced long-lasting modulation of synaptic connectivity

Pyramidal neurons in the piriform cortex from olfactory-discrimination (OD) trained rats undergo synaptic modifications that last for days after learning. A particularly intriguing modification is reduced paired-pulse facilitation (PPF) in the synapses interconnecting these cells; a phenomenon thought to reflect enhanced synaptic release. The molecular machinery underlying this prolonged physiological modulation of synaptic connectivity is yet to be described. We have recently shown that extracellular regulated kinase (ERK) pathway and protein kinase C (PKC) are also required for learning-induced enhancement of intrinsic neuronal excitability. Here we examine whether these signal-transduction cascades are instrumental for the learning-induced, long-lasting PPF reduction. Days after learning completion, PD98059, a selective inhibitor of MEK, the upstream kinase of ERK, increased PPF in neurons from trained, but not in neurons from naïve and pseudo-trained rats. Consequently, the differences in PPF between neurons from trained rats and controls were abolished. The level of activated ERK in synaptoneurosomes was significantly higher in piriform cortex samples prepared from trained rats. Notably, ERK activation revealed that PPF reduction lags behind ERK activation by 2 d. Similarly, the PKC blocker, GF-109203X, enhanced PPF in neurons from trained rats only, thus abolishing the differences between groups. Interestingly, the PKC activator, OAG, had no effect, indicating that PKC activation is required, but not sufficient for long-lasting PPF reduction. Our data show that persistent ERK activation has a key role in maintaining learning-induced PPF reduction for days. This time frame of compartmental ERK-dependent synaptic modulation suggests a novel role for ERK in cortical function.

Olfactory learning prevents MK-801-induced psychosis-like behaviour in an animal model of schizophrenia

There is mounting evidence to support the concept that education is associated with the formation of a functional reserve in the brain, a process that appears to provide some protection against certain aspects of severe central nervous system disorders. The goal of this study was to examine whether learning prevents psychosis-like behaviour in an animal model of schizophrenia. A series of behavioural tasks were used to assess olfactory learning-induced protection against the effects of NMDA channel blocker, MK801. This blocker caused sensory-motor disturbances, spatial learning acquisition deficit, and swimming strategy alterations in pseudo-trained and naive rats, but had a considerably lesser effect on trained rats. In sharp contrast, olfactory learning provided no protection against d-amphetamine application. Our data support the notion that learning-induced protection against schizophrenic behaviour is maintained by non-NMDA-mediated enhanced activation of local connections in the relevant cortical networks.

Fractional Langevin equation: overdamped, underdamped, and critical behaviors

The dynamical phase diagram of the fractional Langevin equation is investigated for a harmonically bound particle. It is shown that critical exponents mark dynamical transitions in the behavior of the system. Four different critical exponents are found. (i) alpha_{c}=0.402+/-0.002 marks a transition to a nonmonotonic underdamped phase, (ii) alpha_{R}=0.441... marks a transition to a resonance phase when an external oscillating field drives the system, and (iii) alpha_{chi_{1}}=0.527... and (iv) alpha_{chi_{2}}=0.707... mark transitions to a double-peak phase of the "loss" when such an oscillating field present. As a physical explanation we present a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing over and underdamped regimes, with or without resonances, show behaviors different from normal.

Random time-scale invariant diffusion and transport coefficients

Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement delta2[over ] of individual particles remains a random variable while indicating that the particle motion is subdiffusive. We investigate this type of ergodicity breaking within the continuous time random walk model and show that delta2[over ] differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable delta2[over ]. Similarly we quantify the response to a constant external field, revealing a generalization of the Einstein relation. Consequences for the interpretation of single molecule tracking data are discussed.

Critical exponent of the fractional Langevin equation

We investigate the dynamical phase diagram of the fractional Langevin equation and show that critical exponents mark dynamical transitions in the behavior of the system. For a free and harmonically bound particle the critical exponent alpha(c)=0.402+/-0.002 marks a transition to a nonmonotonic underdamped phase. The critical exponent alpha(R)=0.441... marks a transition to a resonance phase, when an external oscillating field drives the system. Physically, we explain these behaviors using a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing the underdamped, the overdamped and critical frequencies of the fractional oscillator, recently used to model single protein experiments, show behaviors vastly different from normal.

Distribution of time-averaged observables for weak ergodicity breaking

We find a general formula for the distribution of time-averaged observables for systems modeled according to the subdiffusive continuous time random walk. For Gaussian random walks coupled to a thermal bath we recover ergodicity and Boltzmann's statistics, while for the anomalous subdiffusive case a weakly nonergodic statistical mechanical framework is constructed, which is based on Lévy's generalized central limit theorem. As an example we calculate the distribution of X, the time average of the position of the particle, for unbiased and uniformly biased particles, and show that X exhibits large fluctuations compared with the ensemble average <X>.

Photon statistics for single-molecule nonlinear spectroscopy

We develop the theory of nonlinear spectroscopy for a single molecule undergoing stochastic dynamics and interacting with a sequence of two laser pulses. We find general expressions for the photon counting statistics and the exact solution to the problem for the Kubo-Anderson process. In the limit of impulsive pulses the information on the photon statistics is contained in the molecule's dipole correlation function. The selective limit, the semiclassical approximation, and the fast modulation limit exhibit rich general behaviors of this new type of spectroscopy. We show how the design of external fields leads to insights on ultrafast dynamics of individual molecules that are different from those found for an ensemble.

Upregulation of neurotrophic factors selectively in frontal cortex in response to olfactory discrimination learning

We have previously shown that olfactory discrimination learning is accompanied by several forms of long-term enhancement in synaptic connections between layer II pyramidal neurons selectively in the piriform cortex. This study sought to examine whether the previously demonstrated olfactory-learning-task-induced modifications are preceded by suitable changes in the expression of mRNA for neurotrophic factors and in which brain areas this occurs. Rats were trained to discriminate positive cues in pair of odors for a water reward. The relationship between the learning task and local levels of mRNA for brain-derived neurotrophic factor, tyrosine kinase B, nerve growth factor, and neurotrophin-3 in the frontal cortex, hippocampal subregions, and other regions were assessed 24 hours post olfactory learning. The olfactory discrimination learning activated production of endogenous neurotrophic factors and induced their signal transduction in the frontal cortex, but not in other brain areas. These findings suggest that different brain areas may be preferentially involved in different learning/memory tasks.

Occupation time statistics in the quenched trap model

We investigate the distribution of the occupation time of a particle undergoing a random walk among random energy traps and in the presence of a deterministic potential field. When the distribution of energy traps is exponential with a width T(g), we find in thermal equilibrium a transition between Boltzmann statistics when T>T(g) to Lamperti statistics when T < T(g). We explain why our main results are valid for other models of quenched disorder, and discuss briefly implications on single particle experiments.

Strong correlations between fluctuations and response in aging transport

The correlations between the response of a random walker to an external driving field F switched on at time t_{a}, with the particle's fluctuations in the aging period (0,t_{a}) are investigated. Using the continuous time random walk and the quenched trap model, it is shown that these correlations remain finite even in the asymptotic limit t_{a}-->infinity. Linear response theory gives a relation between the correlations, the fractional diffusion coefficient, and the field F , thus generalizing the Einstein relation. In systems which exhibit aging, fluctuations in the aging period can be used to statistically predict the nonidentical response of particles to an external field.

Aging correlation functions for blinking nanocrystals, and other on-off stochastic processes

Following recent experiments on power law blinking behavior of single nanocrystals, we calculate two-time intensity correlation functions I(t)I(t+t') for these systems. We use a simple two state (on and off) stochastic model to describe the dynamics. We classify possible behaviors of the correlation function and show that aging, e.g., dependence of the correlation function on age of process t, is obtained for classes of the on time and off time distributions relevant to experimental situation. Analytical asymptotic scaling behaviors of the intensity correlation in the double time t and t' domain are obtained. In the scaling limit I(t)I(t+t('))-->h(x), where four classes of behaviors are found: (i) finite averaged on and off times x=t' (standard behavior); (ii) on and off times with identical power law behaviors x=t/t' (case relevant for capped nanocrystals); (iii) exponential on times and power law off times x=tt' (case relevant for uncapped nanocrystals); (iv) for defected off time distribution we also find x=t+t'. Origin of aging behavior is explained based on simple diffusion model. We argue that the diffusion controlled reaction A+B <==>AB, when followed on a single particle level exhibits aging behavior.

Learning in the absence of experience-dependent regulation of NMDAR composition

Olfactory discrimination (OD) learning consists of two phases: an initial N-methyl-D-aspartate (NMDA) receptor-sensitive rule-learning phase, followed by an NMDA receptor (NMDAR)-insensitive pair-learning phase. The rule-learning phase is accompanied by changes in the composition and function of NMDARs at synapses in the piriform cortex, resulting in a high level of the NR2a subunit relative to NR2b. Here we show that the learning-induced changes in NMDAR composition in the adult piriform cortex are due to a decrease in the level of the NR2b subunit protein, rather than an increase in the level of NR2a. Chronic administration of an NMDAR open channel blocker during training delays OD learning and blocks learning-induced changes in NMDAR subunit composition. However, the animals still learn the OD task. Our data demonstrate that learning can occur in the absence of activity-dependent regulation of NMDAR composition, suggesting differences in the mechanism for long-term maintenance of NMDAR-dependent and NMDAR-independent learning.

Photon Counting Statistics for Blinking CdSe−ZnS Quantum Dots: A Lévy Walk Process

We analyze photon statistics of blinking CdSe-ZnS nanocrystals interacting with a continuous wave laser field, showing that the process is described by a ballistic Lévy walk. In particular, we show that Mandel's Q parameter, describing the fluctuations of the photon counts, is increasing with time even in the limit of long time. This behavior is in agreement with the theory of Silbey and co-workers (Jung et al. Chem. Phys. 2002, 284, 181), and in contrast to all existing examples where Q approaches a constant, independent of time in the long time limit. We then analyze the distribution of the time averaged intensities, showing that they exhibit a nonergodic behavior, namely, the time averages remain random even in the limit of a long measurement time. In particular, the distribution of occupation times in the on-state compares favorably to a theory of weak ergodicity breaking of blinking nanocrystals. We show how our data analysis yields information on the amplitudes of power-law decaying on and off time distributions, information not available using standard data analysis of on and off time histograms. Photon statistics reveals fluctuations in the intensity of the bright state indicating that it is composed of several states. Photon statistics exhibits a Lévy walk behavior also when an ensemble of 100 dots is investigated, indicating that the strange kinetics can be observed already at the level of small ensembles.

Learning-induced reversal of the effect of noradrenalin on the postburst AHP

Pyramidal neurons in the piriform cortex from olfactory-discrimination-trained rats have reduced postburst afterhyperpolarization (AHP), for 3 days after learning, and are thus more excitable during this period. Such AHP reduction is caused by decreased conductance of one or more of the calcium-dependent potassium currents, I(AHP) and sI(AHP), that mediate the medium and slow AHPs. In this study, we examined which potassium current is reduced by learning and how the effect of noradrenalin (NE) on neuronal excitability is modified by such reduction. The small conductance (SK) channels inhibitor, apamin, that selectively blocks I(A)(HP), reduced the AHP in neurons from trained, naïve, and pseudotrained rats to a similar extent, thus maintaining the difference in AHP amplitude between neurons from trained rats and controls. In addition, the protein expression level of the SK1, SK2, and SK3 channels was also similar in all groups. NE, which was shown to enhance I(AHP) while suppressing (S)I(AHP), reduced the AHP in neurons from controls but enhanced the AHP in neurons from trained rats. Our data show that learning-induced enhancement of neuronal excitability is not the result of reduction in the I(AHP) current. Thus it is probably mediated by reduction in conductance of the other calcium-dependent potassium current, sI(AHP). Consequently, the effect of NE on neuronal excitability is reversed. We propose that the change in the effect of NE after learning may act to counterbalance learning-induced hyperexcitability and preserve the piriform cortex ability to subserve olfactory learning.

Random walk to a nonergodic equilibrium concept

Random walk models, such as the trap model, continuous time random walks, and comb models, exhibit weak ergodicity breaking, when the average waiting time is infinite. The open question is, what statistical mechanical theory replaces the canonical Boltzmann-Gibbs theory for such systems? In this paper a nonergodic equilibrium concept is investigated, for a continuous time random walk model in a potential field. In particular we show that in the nonergodic phase the distribution of the occupation time of the particle in a finite region of space approaches U- or W-shaped distributions related to the arcsine law. We show that when conditions of detailed balance are applied, these distributions depend on the partition function of the problem, thus establishing a relation between the nonergodic dynamics and canonical statistical mechanics. In the ergodic phase the distribution function of the occupation times approaches a delta function centered on the value predicted based on standard Boltzmann-Gibbs statistics. The relation of our work to single-molecule experiments is briefly discussed.

Learning-induced long-term synaptic modifications in the olfactory cortex

The idea that memory is manifested at the cellular level by enhancement of synaptic connections between simultaneously activated neurons has been suggested half a century ago by Hebb, and is widely accepted since. Much effort is done to describe such enhancement and reveal the underlying mechanisms. Learning-induced synaptic modifications were studied in the last decade with in-vitro brain slices preparations. Several forms of long-term enhancement of synaptic connections between layer II pyramidal neurons in the piriform cortex accompany olfactory learning. Such modifications were described also in other brain areas, following other training paradigms. Post-synaptic enhancement of synaptic transmission is indicated by reduced rise time of (post synaptic potentials) PSPs and formation of new synaptic connections is indicated by increased spine density along dendrites of these neurons. Enhanced synaptic release is indicated by reduced paired-pulse facilitation. In slices from trained rats predisposition for long-term potentiation is decreased and predisposition for long-term depression is increased. These modifications are attributed to olfactory-discrimination rule learning, rather than to memories for specific odors, and may be subsequent to intrinsic modifications in pyramidal neurons that create favorable conditions for activity-dependent synaptic enhancement.

Photon emission from a driven single-molecule source: A renormalization group approach

The photon emission from a single molecule driven simultaneously by a laser and a slow electric radio frequency (rf) field is studied. We use a non-Hermitian Hamiltonian approach which accounts for the radiative decay of a two-level system modeling the single-molecule source. We apply the renormalization group method for differential equations to obtain long time solution of the corresponding Schrodinger equation, which allows us to calculate the average waiting time for the first photon emission. Then, we analyze the conditions for suppression and enhancement of photon emission in this dissipative two-level system. In particular we derive a transcendental equation, which yields the nontrivial rf field control parameters, for which enhancement and suppression of photon emission occurs. For finite values of radiative decay rate an abrupt transition to the state when both situations are indistinguishable is found for certain values of the rf field parameters. Our results are shown to be in agreement with the available experiments [Ch. Brunel et al., Phys. Rev. Lett. 81, 2679 (1998)].

Single-molecule chemical reactions: reexamination of the Kramers approach

Single-molecule chemical reactions yield insight into fluctuation phenomena that are obscured in the measurement of the ensemble of molecules. Kramers escape problem is investigated here in a framework suitable for single-molecule reactions. In particular we obtain distributions of escape times in simple limiting cases, rather than their mean, and investigate their sensitivity on initial conditions. Rich physical behaviors are observed: sub-Poissonian statistics when the dynamics is only slightly deviating from the Newtonian, super-Poissonian behavior when diffusion is dominating, and Poissonian behavior when Kramers original conditions hold. By varying initial conditions escape time distributions can follow a (usual) exponential or a tau(-3/2) decay, due to regular diffusion. We briefly address experimental results that yield the tau(-3/2) behavior (with cutoffs) and propose that this behavior is universal.

Olfactory learning-induced morphological modifications in single dendritic spines of young rats

Learning-related morphological modifications in single dendritic spines were studied quantitatively in the brains of young Sprague-Dawley rats. We have previously shown that olfactory discrimination rule-learning results in transient physiological and morphological modifications in piriform cortex pyramidal neurons. In particular, spine density along the apical dendrites of neurons from trained rats is increased after learning. The aim of the present study was to identify and describe olfactory learning-induced modifications in the morphology of single spines along apical dendrites of the same type of neurons. By using laser-scanning confocal microscopy, we show that 3 days after training completion spines on neurons from olfactory discrimination trained rats are shorter as compared to spines on neurons from control rats. Further analysis revealed that spine shortening attributed to olfactory discrimination learning derives from shortening of spine head and not from shortening of spine neck. In addition, detailed analysis of spine head volume suggests that spines with large heads are absent after learning. As spine head size may be related to the efficacy of the synapse it bears, we suggest that modifications in spine head dimensions following olfactory rule-learning enhance the cortical network ability to enter into a 'learning mode', in which memories of new odours can be acquired rapidly and efficiently.

A Cellular Correlate of Learning-induced Metaplasticity in the Hippocampus

Metaplasticity, the plasticity of synaptic plasticity, is thought to have a pivotal role in activity-dependent modulation of synaptic connectivity, which underlies learning and memory. Metaplasticity is usually attributed to modifications in glutamate receptor-mediated synaptic transmission. However, experimental evidence and theoretical considerations suggest that learning reduces the predisposition for further synaptic strengthening, while behavioral studies show that learning capability is enhanced by prior learning. Here we show that enhanced neuronal excitability in CA1 pyramidal neurons, but not enhanced synaptic transmission, occurs prior to rule learning of an olfactory discrimination task. This transient enhancement lasts for 1 day after rule learning, is apparent throughout the cell population and results from reduction in the medium and slow after-hyperpolarizations that control spike frequency adaptation. Such olfactory learning-induced increased excitability in hippocampal neurons enhances the rats' learning capability in another hippocampus-dependent task, the Morris water maze. Once olfactory discrimination rule learning is acquired, its maintenance is not dependent on the reduced post-burst AHP in hippocampal neurons. However, the enhanced spatial learning capability of olfactory-trained rats in the water maze is diminished once the post burst AHP in CA1 pyramidal cells resumes its initial value. We suggest that enhanced excitability of CA1 neurons may serve as a mechanism for generalized enhancement of hippocampus-dependent learning capability. In the presence of such enhanced neuronal excitability, the hippocampal network enters into a 'learning mode' in which a variety of hippocampus-dependent skills are acquired rapidly and efficiently.

Dynamics of learning-induced cellular modifications in the cortex

This aim of this review is to describe the dynamics of learning-induced cellular modifications in the rat piriform (olfactory) cortex after olfactory discrimination learning and to describe their functional significance to long-term memory consolidation. The first change to occur is in the intrinsic properties of the neurons. One day after learning, pyramidal neurons show enhanced neuronal excitability. This enhancement results from reduction in calcium-dependent conductance that mediates the post burst after-hyperpolarization. Such enhanced excitability lasts for 3 days and is followed by a series of synaptic modifications. Several forms of long-term enhancement in synaptic connections between layer II pyramidal neurons in the piriform cortex accompany olfactory learning. Enhanced synaptic release is indicated by reduced paired-pulse facilitation. Post-synaptic enhancement of synaptic transmission is indicated by reduced rise time of post-synaptic potentials and formation of new synaptic connections is indicated by increased spine density along dendrites of these neurons. Such modifications last for up to 5 days. Thus, olfactory discrimination rule learning is accompanied by a series of cellular modifications which occur and then disappear at different times. These modifications overlap partially, allowing the maintenance of the cortical system in a 'learning mode' in which memories for specific odors can be acquired rapidly and efficiently.

Dynamics of learning-induced spine redistribution along dendrites of pyramidal neurons in rats

We have previously shown that olfactory-discrimination (OD) learning is accompanied by enhanced spine density along proximal apical dendrites of layer II pyramidal neurons in the piriform (olfactory) cortex. Here we studied the temporal dynamics of learning-induced modifications in dendritic spine density throughout the dendritic trees of these neurons. We observed a transient increase in proximal apical spine density after OD learning, suggesting a strengthening of intrinsic excitatory inputs interconnecting neurons within the olfactory cortex. By contrast, the afferent pathway receiving direct input from the olfactory bulb shows spine pruning, suggesting that the connectivity is weakened. The changes in spine density can be attributed to a net change in number of spines, as the morphometric parameters of the dendrites are unaffected by learning. We suggest that spine density changes may represent a mechanism of selective synaptic reorganization required for olfactory learning consolidation.

Nonergodicity of blinking nanocrystals and other Lévy-walk processes

We investigate the nonergodic properties of blinking nanocrystals modeled by a Lévy-walk stochastic process. Using a nonergodic mean field approach we calculate the distribution functions of the time averaged intensity correlation function. We show that these distributions are not delta peaked on the ensemble average correlation function values; instead they are W or U shaped. Beyond blinking nanocrystals our results describe ergodicity breaking in systems modeled by Lévy walks , for example, certain types of chaotic maps and spin dynamics to name a few.

Olfactory learning-induced increase in spine density along the apical dendrites of CA1 hippocampal neurons

We have previously shown that rule learning of an olfactory discrimination task is accompanied by increased spine density along the apical dendrites of piriform cortex pyramidal neurons. The purpose of the present study was to examine whether such olfactory learning task, in which the hippocampus is actively involved, induces morphological modifications in CA1 pyramidal neurons as well. Rats were trained to discriminate positive cues in pairs of odors for a water reward. Morphological modifications were studied in Golgi-impregnated neurons with light microscopy, 1 and 3 days after training completion. Spine densities were measured on the proximal region of apical dendrites and on basal dendrites after rule learning. Three days after training completion, the mean spine density on apical dendrites in neurons from trained rats was significantly higher by 20.5% than in neurons from pseudo-trained and naive animals, which did not differ from each other. By contrast, there was no significant difference in spine density of basal dendrites among the three groups. As length and diameter of spiny dendritic segments did not change after learning, the learning-related increase in spine density in neurons from trained rats may reflect a net increase in the number of excitatory synapses in the hippocampus following olfactory rule learning.

Olfactory learning-related NCAM expression is state, time, and location specific and is correlated with individual learning capabilities

The notion that long-term synaptic plasticity is generated by activity-induced molecular modifications is widely accepted. It is well established that neural cell adhesion molecule (NCAM) is one of the prominent modulators of synaptic plasticity. NCAM can be polysialylated (PSA-NCAM), a reaction that provides it with anti-adhesion properties. In this study we have focused on NCAM and on its polysialylated state, and their relation to learning of an olfactory discrimination task, which depends on both the piriform (olfactory) cortex and hippocampus. We trained rats to distinguish between pairs of odors until rule learning was achieved, a process that normally lasts 6-8 days. At four time points, during training and after training completion, synaptic NCAM and PSA-NCAM expression were assessed in the piriform cortex and hippocampus. We report that NCAM modulation is specific to PSA-NCAM, which is upregulated in the hippocampus one day after training completion. We also report a correlation between the performance of individual rats in an early training stage and their NCAM expression, both in the piriform cortex and hippocampus. Since individual early performance in our odor discrimination task is correlated with the performance throughout the training period, we conclude that early NCAM expression is associated with odor learning capability. We therefore suggest that early synaptic NCAM expression may be one of the factors determining the capability of rats to learn.

Dynamics of olfactory learning-induced up-regulation of L1 in the piriform cortex and hippocampus

L1 is a cell adhesion molecule implicated in the formation of neural circuits and synaptic plasticity. We have examined the sequence and time-frame in which modifications in the synaptic expression of L1 occur in the piriform cortex and hippocampus in the course of rule learning of an olfactory discrimination task. Rats were trained to choose the correct odour in a pair to be rewarded with drinking water. Such training requires 6-8 days on average before rats reach maximal performance. We observed a learning-induced L1 up-regulation that occurred at an early training stage in the piriform cortex but only after rule-learning establishment in the hippocampus. We suggest that the dynamics of L1 up-regulation may reflect the functional role of these brain regions in olfactory rule learning.

Influence of spectral diffusion on single-molecule photon statistics

We investigate the distribution of the number of photons emitted by a single molecule undergoing a spectral diffusion process and interacting with a continuous wave field. Using a generating function formalism an exact analytical formula for Mandel's Q parameter is obtained. The solution, which is valid for weak and strong excitation fields, exhibits transitions between (i) quantum sub-Poissonian and classical super-Poissonian behaviors, and (ii) fast to slow modulation limits.

A Molecular Mechanism for Stabilization of Learning-Induced Synaptic Modifications

Olfaction is a principal sensory modality in rodents, and rats quickly learn to discriminate between odors and to associate odor with reward. Here we show that such olfactory discrimination (OD) learning consists of two phases with distinct cellular mechanisms: an initial NMDAR-sensitive phase in which the animals acquire a successful behavioral strategy (rule learning), followed by an NMDAR-insensitive phase in which the animals learn to distinguish between individual odors (pair learning). Rule learning regulates the composition of synaptic NMDARs in the piriform cortex, resulting in receptors with a higher complement of the NR2a subunit protein relative to NR2b. Rule learning also reduces long-term potentiation (LTP) induced by high-frequency stimulation of the intracortical axons in slices of piriform cortex. As NR2a-containing NMDARs mediate shorter excitatory postsynaptic currents than those containing NR2b, we suggest that learning-induced regulation of NMDAR composition constrains subsequent synaptic plasticity, thereby maintaining the memory encoded by experience.

Lévy statistics for random single-molecule line shapes in a glass

We demonstrate that the statistical behavior of random line shapes of single tetra-tert-butylterrylene chromophores embedded in an amorphous polyisobutylene matrix at T=2 K is described by Lévy statistics as predicted theoretically by Barkai, Silbey, and Zumofen [Phys. Rev. Lett. 84, 5339 (2000)]. This behavior is a manifestation of the long-range interaction between two-level systems in the glass and the single molecule. A universal amplitude ratio is investigated, which shows that the standard tunneling model assumptions are compatible with the experimental data.

Learning-induced reduction in post-burst after-hyperpolarization (AHP) is mediated by activation of PKC

We studied the role of protein kinase C (PKC) and protein kinase A (PKA) in mediating learning-related long lasting reduction of the post-burst after-hyperpolarization (AHP) in cortical pyramidal neurons. We have shown previously that pyramidal neurons in the rat piriform (olfactory) cortex from trained (TR) rats have reduced post-burst AHP for 3 days after odour-discrimination learning, and that this reduction is due to decreased conductance of calcium-dependent potassium current. In the present study, we examined whether this long-lasting reduction in AHP is mediated by second messenger systems. The broad-spectrum kinase inhibitor, H7, increased the AHP in neurons from TR rats, but not in neurons from pseudo-trained (pseudo-TR) and naive rats. Consequently, the difference in AHP amplitude between neurons from TR and control animals was diminished. This effect was also obtained by application of the specific PKC inhibitor, GF-109203x. The PKC activator, 1-Oleoyl-2-acetyl-sn-glycerol (OAG), significantly reduced the AHP in neurons from naive and pseudo-TR rats, but not in neurons from TR rats, so that the difference between the groups was abolished. The PKA-specific inhibitor, H-89, increased the AHP in neurons from all groups to a similar extent, and the difference in AHP amplitude between neurons from TR rats and neurons from controls was maintained. We suggest that while the post-burst AHP in piriform cortex pyramidal neurons is modulated by both PKC and PKA, a PKC-dependent process maintains the learning-related reduction of the AHP in these cells.

Learning-induced enhancement of postsynaptic potentials in pyramidal neurons

We studied the effect of olfactory learning-induced modifications in piriform (olfactory) cortex pyramidal neurons on the propagation of postsynaptic potentials (PSPs). Rats were trained to distinguish between odors in pairs, in an olfactory discrimination task. Three days after training completion, PSPs were evoked in layer II pyramidal cells in piriform cortex brain slices by electrical stimulation of two pathways. Stimulation of layer Ib activated the intra-cortical fibers that terminate on the proximal region of the apical and basal dendrites. Stimulation of layer Ia activated the afferent axons that originate from the olfactory bulb and terminate on the distal apical dendrites. We have previously shown that olfactory training is accompanied by enhanced synaptic transmission in the intrinsic pathway, but not in the afferent pathway at 3 days after training. Here we show that at this stage, in both pathways PSPs evoked in neurons from trained rats had significantly faster rise time measured at the soma compared with PSPs in neurons from pseudo-trained and naive rats. Activation of the slow afterhyperpolarization (AHP), which is generated by potassium channels probably located at the proximal region of both apical and basal dendrites, reduced the amplitude measured at the soma of the proximal intrinsic pathway PSPs more effectively than PSPs that were generated distally by the afferent fibers. Thus the amount of reduction by AHP was used as a measure for the relative distance of PSP-generating sites from the soma. In neurons from trained rats, despite the previously reported reduction in AHP amplitude, AHP conductance shunted the PSPs from both synaptic pathways more efficiently compared with neurons from the control rats. We suggest that in neurons from trained rats PSPs are electrotonicly closer to the soma.

Time-dependent fluctuations in single molecule spectroscopy: a generalized Wiener-Khintchine approach

We consider the theory of time-dependent fluctuations in single molecule spectroscopy. In particular, the relation between stationary spectral diffusion processes and photon counting statistics is investigated. Within linear response theory, a general relation between Mandel's Q parameter and a three time dipole correlation function is obtained. For a prototype spectral diffusion process, an exact solution of Q is found which exhibits rich types of physical behaviors.

Cellular correlates of olfactory learning in the rat piriform cortex

This review describes research that combines cellular physiology with behavioral neuroscience, to study the cellular mechanisms underlying learning and memory in the mammalian brain. Rats were trained with an olfactory conditioning paradigm, in which they had to memorize odors in order to be rewarded with drinking water. Such training results in rule learning, which enables enhanced acquisition of odor memory. Training results in the following learning-related physiological modifications in intrinsic and synaptic properties in olfactory (piriform) cortex pyramidal neurons: 1. increased neuronal excitability, indicated by reduced afterhyperpolarization, and 2. increased synaptic transmission, indicated by reduced paired-pulse facilitation. These modifications are correlated to enhanced learning capability rather than to storage of memory for specific odors. In addition, using a different paradigm of odor-training, it is shown that NMDA and betra-adrenergic receptors are involved at different stages of long-term memory consolidation.

Olfactory Learning Modifies Predisposition for Long-term Potentiation and Long-term Depression Induction in the Rat Piriform (Olfactory) Cortex

Learning-related modifications in predisposition for long-term potentiation (LTP) and long-term depression (LTD) were studied in brain slices of the rat piriform cortex following olfactory learning. Rats were trained to discriminate between pairs of odors until they demonstrated rule learning. We have previously shown that such training is accompanied by enhanced neuronal excitability and increased synaptic transmission in the intrinsic synaptic pathway. Here we show that the susceptibility for further enhancing synaptic connectivity by inducing LTP in slices from trained rats is markedly reduced after training, compared with slices from pseudo-trained and naive rats. Accordingly, while 900 stimuli at 1 Hz did not induce LTD in slices from control rats, it induced significant LTD in slices from trained rats. Post-tetanic potentiation (PTP) was also reduced after training, indicating that synaptic release is enhanced after odor learning, as previously suggested. We suggest that learning-related cellular modifications and activity-dependent synaptic plasticity share a common mechanism in the primary olfactory cortex. Our data also support the prediction generated according to the sliding modification threshold theory that learning should be accompanied by reduced capability of inducing LTP and increased susceptibility for LTD induction.

Fractional Fokker-Planck equation, solution, and application

Recently, Metzler et al. [Phys. Rev. Lett. 82, 3563 (1999)], introduced a fractional Fokker-Planck equation (FFPE) describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field, and a Boltzmann thermal heat bath. In this paper we present the solution of the FFPE in terms of an integral transformation. The transformation maps the solution of ordinary Fokker-Planck equation onto the solution of the FFPE, and is based on Lévy's generalized central limit theorem. The meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk (CTRW). We investigate in detail (i) a force-free particle, (ii) a particle in a uniform field, and (iii) a particle in a harmonic field. We also find an exact solution of the CTRW, and compare the CTRW result with the corresponding solution of the FFPE. The relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given. An example of the one-dimensional fractional first passage time in an external linear field is investigated in detail. The FFPE is shown to be compatible with the Scher-Montroll approach for dispersive transport, and thus is applicable in a large variety of disordered systems. The simple FFPE approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena.

Long-lasting cholinergic modulation underlies rule learning in rats

We studied the role of acetylcholine (ACh) in creating learning-related long-lasting modifications in the rat cortex. Rats were trained to discriminate positive and negative cues in pairs of odors, until they demonstrated rule learning and entered a mode of high capability for learning of additional odors. We have previously reported that pyramidal neurons in olfactory (piriform) cortex from trained rats had reduced spike afterhyperpolarization (AHP) for 3 d after rule learning. In the present study we examined the mechanism underlying this long-lasting modification. The cholinergic agonist carbachol reduced both slow AHP and firing adaptation in neurons from pseudotrained rats, but had no effect on neurons from trained rats, suggesting pre-existing cholinergic effect. Intracellular application of the calcium chelator BAPTA abolished the difference in slow AHP and in adaptation between groups, suggesting that the difference resulted from reduction in the ACh-sensitive, Ca(2+)-dependent potassium current, I(AHP). At the behavioral level, application of the muscarinic blocker scopolamine before each training session delayed rule learning but had no effect on further acquisition of odor memory. We suggest that intense ACh activity during rule learning enhances neuronal excitability in the piriform cortex by reducing I(AHP) and that the effect outlasts the stage of rule learning, so that ACh activity is not crucial for further odor learning.

Olfactory learning is associated with increased spine density along apical dendrites of pyramidal neurons in the rat piriform cortex

We studied the effect of olfactory learning on the dendritic spine density of pyramidal neurons in the rat piriform (olfactory) cortex. Rats were trained to distinguish between two pairs of odours in an olfactory discrimination task. Three days after training completion, rats were killed and layer II pyramidal neurons identified by Golgi impregnation were examined with a light microscope. Counts of visible spines were performed along the secondary and tertiary branches of both the apical dendrites and the basal dendrites, which are the sites of intracortical synaptic inputs. An estimate of the true spine density was obtained using Feldman and Peters' method (1979, The Journal of Comparative Neurology, 188, 527--542). The estimated true spine density along apical dendrites was higher in neurons from trained rats than those in pseudotrained and naive rats by 15%. As length of spiny dendrites did not change significantly after learning, the learning-related increase in spine density in neurons from trained rats may indicate on an increased number of excitatory synapses interconnecting pyramidal neurons in the piriform cortex, following olfactory learning.

Levy distribution of single molecule line shape cumulants in glasses

We investigate the distribution of single molecule line shape cumulants, kappa(1),kappa(2),ellipsis, in low temperature glasses based on the sudden jump, standard tunneling model. We find that the cumulants are described by Levy stable laws, thus the generalized central limit theorem is applicable for this problem.

One-dimensional stochastic Levy-lorentz gas

We introduce a Levy-Lorentz gas in which a light particle is scattered by static point scatterers arranged on a line. We investigate the case where the intervals between scatterers xi(i) are independent random variables identically distributed according to the probability density function &mgr;(xi) approximately xi(-(1+gamma)). We show that under certain conditions the mean square displacement of the particle obeys <x(2)(t)>>/=Ct3-gamma for 1<gamma<2. This behavior is compatible with a renewal Levy walk scheme. We discuss the importance of rare events in the proper characterization of the diffusion process.

From continuous time random walks to the fractional fokker-planck equation

We generalize the continuous time random walk (CTRW) to include the effect of space dependent jump probabilities. When the mean waiting time diverges we derive a fractional Fokker-Planck equation (FFPE). This equation describes anomalous diffusion in an external force field and close to thermal equilibrium. We discuss the domain of validity of the fractional kinetic equation. For the force free case we compare between the CTRW solution and that of the FFPE.

Reduced synaptic facilitation between pyramidal neurons in the piriform cortex after odor learning

Learning-related cellular modifications were studied in the rat piriform cortex after operand conditioning. Rats were trained to discriminate positive cues in pairs of odors. In one experimental paradigm, rats were trained to memorize 35-50 pairs of odors ("extensive training"). In another paradigm, training was continued only until rats acquired the rule of the task, usually after learning the first two pairs of odors ("short training"). "Pseudotrained" and "naive" rats served as controls. We have previously shown that "rule learning" of this task was accompanied by reduced spike afterhyperpolarization in pyramidal neurons in brain slices of the piriform cortex. In the present study, synaptic inputs to the same cells were examined. Pairs of electrical stimuli applied to the intrinsic fibers that interconnect layer II pyramidal neurons revealed significant reduction in paired-pulse facilitation (PPF) in this pathway even after short training. PPF in shortly trained rats was reduced to the same extent as in extensively trained rats. PPF reduction did not result from modification of membrane properties in the postsynaptic cells, change in postsynaptic inhibition, or impairment of the facilitation mechanism. Extracellular field potential recordings showed enhanced synaptic transmission in these synapses. The reduction in PPF became apparent only 3 d after task acquisition and returned to control value 5 d later. PPF evoked by stimulating the afferent fibers to the same neurons was increased 1 d after training for 2 d. We suggest that the transient enhancement in connectivity in the intrinsic pathway is related to the enhanced learning capability and not to memory for specific odors, which lasts for weeks.