You will be presented with the abstract, introduction, and discussion section from a biomedical research article. Your task is to create a summary that a layperson can understand, capturing the essence of the research article. Keep essential scientific terms, but ensure that the language remains clear and the concepts are explained in an uncomplicated manner.

Title: Neuronal Spike Timing Adaptation Described with a Fractional Leaky Integrate-and-Fire Model}
Article:
Abstract:
The voltage trace of neuronal activities can follow multiple timescale dynamics that arise from correlated membrane conductances. Such processes can result in power-law behavior in which the membrane voltage cannot be characterized with a single time constant. The emergent effect of these membrane correlations is a non-Markovian process that can be modeled with a fractional derivative. A fractional derivative is a non-local process in which the value of the variable is determined by integrating a temporal weighted voltage trace, also called the memory trace. Here we developed and analyzed a fractional leaky integrate-and-fire model in which the exponent of the fractional derivative can vary from 0 to 1, with 1 representing the normal derivative. As the exponent of the fractional derivative decreases, the weights of the voltage trace increase. Thus, the value of the voltage is increasingly correlated with the trajectory of the voltage in the past. By varying only the fractional exponent, our model can reproduce upward and downward spike adaptations found experimentally in neocortical pyramidal cells and tectal neurons in vitro. The model also produces spikes with longer first-spike latency and high inter-spike variability with power-law distribution. We further analyze spike adaptation and the responses to noisy and oscillatory input. The fractional model generates reliable spike patterns in response to noisy input. Overall, the spiking activity of the fractional leaky integrate-and-fire model deviates from the spiking activity of the Markovian model and reflects the temporal accumulated intrinsic membrane dynamics that affect the response of the neuron to external stimulation.
Introduction:
The leaky integrator properties of a neuron are determined by the membrane resistance and capacitance which define a single time constant for the membrane voltage dynamics[1]–[3]. However, the voltage trace of real neurons can follow multiple timescale dynamics[4]–[6] that arise from the interaction of multiple active membrane conductances[7]–[11]. Such processes can result in power-law behavior in which the membrane voltage cannot be characterized with a single time constant[5],[12]–[15]. Since power-law dynamics can span all the scales of interest of neuronal behavior[16]–[20], it is necessary to develop a framework to study such processes and their effect on the electrical and computational capacities of neurons. In the classical leaky integrate-and-fire model the temporal evolution of the voltage is local[21],[22]. The value of the voltage at a given time depends only on the value of the voltage in the immediate previous time step. Such a process is called Markovian. However, coupling of active conductances does not allow the value of the voltage to be memoryless[11],[17],[23]–[26]. Instead, long time correlations affect the membrane voltage for hundreds of milliseconds. The emergent effect of these membrane correlations is a non-Markovian process that can be modeled with a fractional derivative[27]–[31]. A fractional derivative represents a non-local process[32]–[34] in which the value of the variable is determined by integrating a temporal weighted voltage trace, also called the memory trace. Although fractional derivatives and integrals are almost as old as traditional calculus[32],[35],[36], they have not been widely used due to limited computer power. In the fractional integrate-and-fire model the exponent of the fractional derivative goes from 0 to 1, with 1 representing the normal derivative. As the exponent of the fractional derivative decreases, the weights of the voltage trace increase. Thus, the value of the voltage is increasingly correlated with the trajectory of the voltage in the past. We developed and analyzed a fractional leaky integrate-and-fire model. The only parameters of the model are the conductance, capacitance, and the fractional exponent. By varying the fractional exponent our model reproduces the upward and downward spike-frequency adaptations found experimentally in pyramidal neurons[37],[38], tectal neurons[39] and fast-spiking cells of layer IV in the rat barrel cortex[40]. Furthermore, the model replicates not only the adapting firing rate but also the long first-spike latency seen in pyramidal neurons in layer V[38]. The model also produces spikes with longer first-spike latency and high inter-spike variability with power-law distribution, which cannot be reproduced by the classical integrate-and-fire model. We further analyze spike adaptation and the responses to noisy and oscillatory input. Overall, the spiking activity of the fractional integrate-and-fire model deviates from the spiking activity of the Markovian model and reflects the temporal accumulated intrinsic membrane dynamics that affect the response of the neuron to external stimulation.
Discussion:
The fractional model can produce different degrees of adapting electrical activities by modifying the fractional exponent. A fractional order derivative captures the long-range correlations of the a system models that results in non-local dynamics. Fractional differential equations have been used in biological systems to capture the long-term memory effects of the dynamics[33],[74],[76]–[81]. For example, fractional order derivatives have been observed in the vestibular-ocular system[27],[82] and in the gating dynamics of ion channels[44],[83]. Power law statistical distributions of a neuronal response also exhibit fractional order dynamics[37]. The voltage dynamics in the fractional order model depend on both the Markov term( immediate past) and the voltage-memory trace that integrates all past voltage values. The behavior of the voltage-memory trace is similar to the behavior of the adaptive filter in the work of Pozzorini et al.[59], although in their work this filter is described as the sum of the spike-triggered current and a moving threshold. The voltage-memory trace also corresponds to the adaptation integral used in other works[18]. In this context our fractional order leaky integrate-and-fire model is a unified mathematical and computational framework that can be used to describe power-law dynamics and long-range correlations in neuronal activity. The fractional derivative can capture relationships between the distribution of conductances that can be complicated to model using explicit techniques. In biophysical systems, the voltage-memory trace might represent spike-triggered mechanisms that cause adaptation. For example, the voltage-memory trace might represent the slowly inactivating potassium-like current that induces the upward spike adaptation shown in Layer V pyramidal neurons of primary motor cortex[38], suggesting that these neurons are fractional differentiators. Alternatively, the voltage-memory trace can correspond to other adaptation currents such as calcium-activated after-hyperpolarization currents[5], slow sodium-channel inactivation currents[84],[85], or a combination of several adaptation currents. Many studies have used models with slow adaptation currents and exponential functions to analyze spike time adaptation[5],[54],[84]. Some of these models can produce similar properties found in fractional dynamics. For example, the Hodgkin-Huxley model with slow after hyper-polarization currents can produce multiple timescale adaptation processes[37]. The Generalized leaky integrate-and-fire model with an adaptive filter( GLIF-) produces spike adaptation with power-law dynamics[59]. Both the power-law dynamics and history-dependent properties of the GLIF- model correspond to that of the fractional leaky integrate-and-fire model. However, the fractional model provides a general way by simplifying the complicated details shown in other models. The fractional model exhibits spike adaptation with power-law dynamics by integrating all the past voltage values without any additional adaptation currents. Power law functions generalize the mechanism underlying exponential processes and are better alternatives to describe scale invariant spike adaptation[17],[18],[42],[59],[86],[87]. The fractional model shows new directions for studying spike adaptation using fractional derivatives and power-law dynamics instead of classical derivatives and exponential functions. The key parameter in our fractional model is. Experimental results have suggested that can be as small as 0. 15 for neocortical pyramidal neurons[37]. Our fitting of these data also resulted in a value of. Fits to the response of Layer 5 pyramidal motor cortex neurons resulted in a value of[38]. Thus, the experimental results and our modeling analysis suggest that the biophysically important values of are when. However, it is clear that if is much closer to 0, the system takes a longer time to generate spikes or never fires spikes at all, depending on the magnitude of the stimulus, so the feasible range of the fractional exponent might be between 0. 05 and 0. 2. The value of might correspond to the type, function, or location of specific neurons[38]. These regional differences are not exclusive to the cortex, for example, Purkinje cells in Lobule X and Lobules III – V show different degrees of spike adaptation[75]. Thus, different values of can be used to map the general voltage and spike time adaptation properties of neurons throughout the brain. Previous work has provided an experimental foundation to determine if a spiking neuron is a fractional differentiator[37]. Our framework provides a more general methodology to determine power-law neuronal dynamics from the sub-threshold to the spiking regime. At the sub-threshold level there are several measurements that can indicate that the membrane of the neuron follows a power-law. For example: In the same experiment a series of protocols can also be applied to determine if the spiking activity follows power-law dynamics. Some of these measurements are straightforward, others require longer recordings. For instance: Although power-laws are found at multiple scales of biological organization, their function and importance are still debated[88],[89]. In our work we propose that the membrane voltage of neurons can follow a power-law due to the emergent property of the combination of multiple active conductances. The value of the fractional derivative can be mapped to spike time adaptation dynamics taking place in multiple cell types across the brain. Computationally, a low value of results in spiking dynamics that are at the same time highly adaptable and reliable. Thus, neurons following power-law adaptation could have a large operational range while providing the reliability to filter out noisy signals and increase information capacity. The lack of a sub-threshold resonance frequency allows the neuron to filter signals homogeneously over a wide range of frequencies. In such a case, the fractional leaky integrate-and-fire model provides the basis to study the computational capacities and information processing properties of neurons showing high degree of spike time adaptation.
Spike adaptation is a property of most neurons . When spike time adaptation occurs over multiple time scales , the dynamics can be described by a power-law . We study the computational properties of a leaky integrate-and-fire model with power-law adaptation . Instead of explicitly modeling the adaptation process by the contribution of slowly changing conductances , we use a fractional temporal derivative framework . The exponent of the fractional derivative represents the degree of adaptation of the membrane voltage , where 1 is the normal leaky integrator while values less than 1 produce increasing correlations in the voltage trace . The temporal correlation is interpreted as a memory trace that depends on the value of the fractional derivative . We identify the memory trace in the fractional model as the sum of the instantaneous differentiation weighted by a function that depends on the fractional exponent , and it provides non-local information to the incoming stimulus . The spiking dynamics of the fractional leaky integrate-and-fire model show memory dependence that can result in downward or upward spike adaptation . Our model provides a framework for understanding how long-range membrane voltage correlations affect spiking dynamics and information integration in neurons .