History-Dependent Excitability as a Single-Cell Substrate of Transient Memory for Information Discrimination

Neurons react differently to incoming stimuli depending upon their previous history of stimulation. This property can be considered as a single-cell substrate for transient memory, or context-dependent information processing: depending upon the current context that the neuron “sees” through the subset of the network impinging on it in the immediate past, the same synaptic event can evoke a postsynaptic spike or just a subthreshold depolarization. We propose a formal definition of History-Dependent Excitability (HDE) as a measure of the propensity to firing in any moment in time, linking the subthreshold history-dependent dynamics with spike generation. This definition allows the quantitative assessment of the intrinsic memory for different single-neuron dynamics and input statistics. We illustrate the concept of HDE by considering two general dynamical mechanisms: the passive behavior of an Integrate and Fire (IF) neuron, and the inductive behavior of a Generalized Integrate and Fire (GIF) neuron with subthreshold damped oscillations. This framework allows us to characterize the sensitivity of different model neurons to the detailed temporal structure of incoming stimuli. While a neuron with intrinsic oscillations discriminates equally well between input trains with the same or different frequency, a passive neuron discriminates better between inputs with different frequencies. This suggests that passive neurons are better suited to rate-based computation, while neurons with subthreshold oscillations are advantageous in a temporal coding scheme. We also address the influence of intrinsic properties in single-cell processing as a function of input statistics, and show that intrinsic oscillations enhance discrimination sensitivity at high input rates. Finally, we discuss how the recognition of these cell-specific discrimination properties might further our understanding of neuronal network computations and their relationships to the distribution and functional connectivity of different neuronal types.


IF model
In the absence of synaptic input (I syn = 0), the solution to (9) reads where v 0 is the initial condition of the voltage variable at time t = 0. Hence, in the absence of post synaptic spikes (see Methods), the response of the model neuron to a train of synaptic events at times (t s 0 , t s 1 , . . . , t s n−1 ) can be calculated by iterating the following map: for i = 1, . . . , n − 1, where ISI i = t s i − t s i−1 , A is the synaptic strength (assumed for simplicity to be equal among synaptic events), and v i is the voltage variable just before the arrival of the synaptic event at time t s i . The voltage variable after the last spike of the train is calculated as v n = (v n−1 + A) and will be used to calculate the discriminability (see the next section). In all our simulations we set the initial condition for the v variables at the rest state (v 0 = 0).

GIF model
In the absence of synaptic input (I syn = 0), the subthreshold dynamics (10) has as a general solution . The parameter µ determines the membrane time constant, while the parameter ω is the oscillation frequency (in 1/rad), related to the intrinsic period of oscillations T = 2π ω . We chose to fix µ at a physiological value (µ = 1). With this choice, ω = √ β. The coefficients c 1 and c 2 depends on the initial conditions and can be easily derived analytically.
Considering that u 1 = 1 + λ 1 1 and u 2 = 1 + λ 2 1 are eigenvectors of the system (10) associated with the eigenvalues λ 1 = −µ + iω and λ 2 = −µ − iω, respectively, by evaluating (A-1) at t = 0 and projecting on the v and w axis one obtains: where v 0 and w 0 are the general initial conditions for the voltage and the slow variable at t = 0. By solving for c 1 and c 2 and substituting e iω = cos(ω) + i sin(ω) one obtains the following expressions for the free evolution of the system (10) given the initial condition (v 0 , w 0 ): Hence, if we model the EPSPs as instantaneous pulses along the v axis, it is possible to compute analytically the response to a train of presynaptic pulses. In the absence of post synaptic spikes (see Methods), the response of the model neuron to a train of synaptic events at times (t s 0 , t s 1 , . . . , t s n−1 ) can be calculated by iterating the following map: A is the synaptic strength (assumed to be equal among synaptic events), and (v i , w i ) are the dynamical variables just before the arrival of the synaptic event at time t s i . The voltage variable after the last spike of the train is calculated as and will be used to calculate the discriminability (see the next section). In all our simulations we set the initial conditions at the rest state (v 0 = 0, w 0 = 0).

Analytical derivation of the discriminability
We define the neuronal excitability in any moment in time as the minimal synaptic strength capable of firing the neuron. The instantaneous intrinsic discriminability between two input histories is then defined, in any moment in time, as the square difference between the excitabilities corresponding to the two input histories. That is, where the last equality is due to the fact that in these linear models the firing threshold depends upon the voltage variable v only. The main measures of intrinsic discriminability we considered are the maximal instantaneous discriminability (maximal value of (A-2)) and the cumulative discriminability, defined as the integral in time from zero to +∞ of the instantaneous discriminability: The instantaneous and the cumulative discriminability can be computed analytically for the IF and GIF neuron, given the initial conditions for the dynamical variables.

Analytical derivation of the discriminability: IF neuron
For the IF model neuron the instantaneous discriminability can be written as a function of the initial conditions for the voltage variable v i 0 and v j 0 , corresponding to the neuron's state after input history H i and H j , respectively: This expression integrated from zero to +∞ gives the cumulative discriminability: Analytical derivation of the discriminability: GIF neuron For the GIF model neuron the instantaneous discriminability can be written as a function of the initial conditions for the dynamical variables (v i 0 , w i 0 ) and (v j 0 , w j 0 ), corresponding to the neuron's state after input history H i and H j , respectively: Calculating the square product of the bracketed expression and applying some trigonometric equalities one obtains: (2ωt) and finally The above expression integrated from zero to +∞ gives the cumulative discriminability: where the indefinite integrals e ax cos bx = b a 2 + b 2 e ax sin bx + a a 2 + b 2 e ax cos bx e ax sin bx = a a 2 + b 2 e ax sin bx − b a 2 + b 2 e ax cos bx have been used. Thus, it is a quadratic function in ∆v = v j − v i and ∆w = w j − w i , whose orientation and eccentricity are determined by the neuron's parameters µ and ω.