Firing map of an almost periodic input function

In mathematical biology and the theory of electric networks the firing map of an integrate-and-fire system is a notion of importance. In order to prove useful properties of this map authors of previous papers assumed that the stimulus function f of the system \dot{x}= f(t,x) is continuous and usually periodic in the time variable. In this work we show that the required properties of the firing map for the simplified model \dot{x}=f(t) still hold if f \in L_{loc}^1(R) and f is an almost periodic function. Moreover, in this way we prepare a formal framework for next study of a discrete dynamics of the firing map arising from almost periodic stimulus that gives information on consecutive resets (spikes).


Introduction
The integrate-and-fire models are used mainly in neuroscience to describe nerve-membrane voltage response to a given input. In these, usually one-dimensional models,ẋ = f (t, x), f : the continuous dynamics govern by a differential equation is interrupted by threshold -reset behavior which is supposed to mimic spiking in real neurons. Precisely, the system evolves from an initial condition according to the differential equation as long as the threshold-value of a dynamical variable is achieved. Then there is an immediate resetting to the resting-value and the dynamics continues from the new initial condition. The resting and threshold values are typically set to 0 and 1, respectively. The sequence of consecutive spikes can be described by the iterates of the firing map. So far an analytical approach was carried mainly towards models with periodic input, where the firing map is a lift of a degree one circle map and the tools of rotation theory can be used ( [4], [5], [8]). Properties of the firing map were also investigated with the help of numerical simulations ( [6], [10]). In analytical survey it was always assumed that the input function is smooth enough. In this work we study the case when the function on the right hand side of a differential equation is in general not continuous and not periodic but almost periodic.
A natural domain of Φ is the set D Φ = {t ∈ R : ∃ s>t x(s; t, 0) = 1}. The mapping Φ assigns to each t ∈ D Φ the value Φ(t), equal to the time after which a trajectory starting from x = 0 at the moment t reaches the line x = 1 for the first time. This is the time at which we have the firing (alternatively called also a spike). Immediately after that the solution x(t) is "reset" to zero and next the system given by (1) evolves from new starting point (Φ(t), 0) until a moment Φ 2 (t) satisfying x(Φ 2 (t); Φ(t), 0) = 1. Thus the iterates Φ n (t) of the firing map can be defined recursively as The constant functions x r = 0 and x τ = 1 are called the reset and threshold functions respectively. One can investigate varying threshold and reset functions (e.g. [8]) but the general cases reduces to the above ( [4]).

Perfect Integrator Model
Consider the system called Perfect Integrator Model (after [4]), i.e. the scalar with "resetting mechanism". Throughout the rest of this paper we assume that f ∈ L 1 loc (R), unless some additional assumptions are stated, and write simply b a f (u) du meaning the Lebesque integral A f (u) du for A = [a, b]. We will denote x(t; t 0 , x 0 ) := t t 0 f (u) du + x 0 and call x(·; t 0 , x 0 ) the solution of (2) (satisfying the initial condition (t 0 , x 0 )) or a trajectory of (2). A locally integrable function has desired properties: the function F (t) = t t 0 f (u) du is a continuous function F : (t 0 , ∞) → R and it assures that the system (2) produces only a finite number of spikes in any finite interval, as we will see after Corollary 1. Moreover, this is a significantly larger class of functions than was studied before ( [4], [5], [6], [8], [10]): it includes, for instance, non-continuous piece-wise constant stimulus functions (e.g. the Haar wavelets), often used in neuroscience (see e.g [9] and references therein).
Since every two trajectories of (2) differ by a constant and the solution x(t; a, x 0 ) = t a f (u) du + x 0 is continuous for f ∈ L 1 loc (R), we get the following loc (R), then for the equation (2) the consecutive iterates of the firing map are equal to Φ n (t) = min{s > t : x(s; t, 0) = n} (3) and there is only a finite number of firings in every bounded interval.
Now we state necessary and sufficient conditions under which the firing map Φ : R → R is properly defined, it is when D Φ = R. In this case Φ induces a semi-dynamical system on R.
In particular, for any t 0 ∈ R (4) is the necessary and sufficient condition for all the iterations {Φ n (t 0 )} ∞ n=1 to be finite real valued.
Proof. Choose t 0 ∈ R. We want all the iterations {Φ(t 0 ), Φ 2 (t 0 ), Φ 3 (t 0 ), ...} to be well-defined according to (3), that is we demand that for all n ≥ 1 there exists s > Φ n−1 (t 0 ) such that x(s) = n. However, this is equivalent to demanding that the solution x : [t 0 , ∞) → R with x(t 0 ) = 0 is unbounded from above. But Obviously, lim sup t→∞ t 0 f (s)ds = ∞ asserts that all the solutions of (2) are unbounded from above. ✷ Throughout the rest of the paper we assume that (4) is satisfied. By using a property of the Lebesque integral which says that if the integral A f (t)dt of a non-negative function on a measurable set A ⊂ R, µ(A) > 0, is positive then there exist t 1 < t 2 such that Lemma 3 Let f ∈ L 1 loc (R) and Φ : R → R be the firing map associated with (2). Then Φ is increasing, correspondingly, non-decreasing, iff f (t) > 0, or f (t) ≥ 0 respectively, a.e. in R.
Proof. The proof that f (t) ≥ 0 (f (t) > 0) a.e. induces non-decreasing (increasing) firing map Φ is almost immediate from the previous lemma. Conversely, suppose that there exists a non-zero measure set A such that of such n is ensured by the fact that there can be only a finite number of firings in the interval [t 1 , t 2 ]). Then, as before, we obtain that Analogously we show that when Φ is strictly increasing, then f is positive, perhaps with the exception of some zero-measure set. ✷ Some general results concerning continuity and injectivity of the firing map for the models of the typeẋ = f (t, x) when f is smooth enough are given in [5]. However, for the Perfect Integrator we obtain a detailed description of continuity if only f ∈ L 1 loc (R).
. Moreover such points are the only points of discontinuity of Φ.
Proof. For the first part of the statement take a sequence t n → t 0 , t n < t 0 . Then To prove the second part of the statement let us take t n →ā, t n >ā. We have already showed that It follows that Φ(ā + ) ≥ Φ(ā) + δ 0 and consequently Φ is not right continuous at a.
As a direct consequence of Proposition 1 we get the following: Corollary 2 If f ∈ L 1 loc (R) and f (t) > 0 almost everywhere, then Φ is continuous.

Firing rate
Definition 2 For every t ∈ R one defines the firing rate, denoted by r(t), as In general for an equationẋ = f (t, x) the limit of (5) might not exist for some t. Moreover, even if the limit exists for all t, it might depend on t. However, for the simplified model (2) we have the following theorem, which was proved in [4] and the proof is valid for f ∈ L 1 loc (R): Theorem 1 Suppose that for the model (2) there exists a finite limit Then for every point t 0 ∈ R the firing rate r(t 0 ) exists and is given by the formula (6). In particular the firing rate r(t) does not depend on t.

Almost periodic functions
In next we consider the model (2) with the function f : R → R almost periodic. There are various classes of almost periodic functions, e.g. in the sense of Bohr, Stepanov, Weyl and Besicovitch. They are defined as the closure of the space of generalized trigonometric polynomials under corresponding norms. The notion of almost periodicity generally holds for functions f : R → C. However, we will only consider almost periodic functions taking real values.

Definition 3
A continuous function f : R → C is almost periodic function in the sense of Bohr (alternatively called also uniformly almost periodic) if for any ε > 0 the set E{ε, f (t)} of all the numbers τ such that for all t ∈ R |f (t + τ ) − f (t)| < ε is relatively dense, i.e. there exists l ε > 0 such that in any interval of length l ε there is at least one τ satisfying this inequality.
The set E{ε, f (t)} is called the set of ε-almost periods of f .
Equivalently we can say that f is Bohr almost periodic if it is the limit of a sequence of generalized trigonometric polynomials P n (t) = k(n) i=1 c i exp(ıλ i t), c i ∈ C and λ i ∈ R, under the norm f = sup t∈R |f (t)|. Any Bohr almost periodic function is uniformly continuous and bounded ( [3], [7]).

Definition 4 A function
The set SE{ε, f (t)} is called the set of ε-Stepanov almost periods of f .
The space of Stepanov almost periodic functions can be obtained also as the closure of generalized trigonometric polynomials under the norm · St.,r,p . Any Stepanov almost periodic function which is uniformly continuous is Bohr almost periodic. Different values of r in · St.,r,p give obviously different norms but the same topology. Thus we will consider Stepanov almost periodic functions with r = 1. If f is a p 1 -Stepanov almost periodic function, then it also p 2 -Stepanov almost periodic for p 2 < p 1 with fixed r ([1], [3]). One can also consider almost periodic functions in the sense of Weyl or Besicovitch, where the class of Besicovitch almost periodic functions with p = 1 is larger and includes all the other mentioned types of almost periodicity. For an almost periodic function f : R → C of any of these types there exists a finite mean ([1], [3], [7]):

From this we conclude that:
Corollary 3 For the model (2), where f : R → R is almost periodic in any of the above sense the firing rate exists and equals the mean of a function, provided that the mean is nonnegative (otherwise the firing rate is zero).
Note that if M{f (t)} < 0, then (4) does not hold and there is no firing.
Definition 5 A function f : R → C is limit-periodic (in the sense of Bohr) if it is a limit of uniformly converging sequence {f n } of continuous periodic functions.
Note that generalized trigonometric polynomials are in general not periodic (consider, for example, P (t) = sin( √ 2t) + sin(2t)) and that a limit-periodic function is an uniformly almost periodic function. One can read in [1] in details about the theory of almost periodic functions. We will write shortly that f is u.a.p. or S.a.p., meaning uniform (Bohr) or Stepanov (with r = 1 and p = 1) almost periodicity, respectively. The mean M is continuous [3], meaning that M{f n } → M{f } if f n → f in any of the "almost-periodic norms" (i.e. the norms with respect to which a given space of almost periodic functions can be obtained as the closure of the space of generalized trigonometric polynomials). Thus the following holds: n=1 be a sequence of almost-periodic functions f n : R → R of the Bohr, Stepanov, Weyl or Besicovitch type converging to f in uniform, Stepanov, Weyl or Besicovitch norm, respectively. Let r n be the firing rate of the equationẋ = f n (t), for n = 1, 2, 3, ..., and r the firing rate for (2).
Then r = lim n→∞ r n .
We end this section with propositions showing when the existence of the positive mean value of f is a necessary and sufficient condition for existence of the firing map Φ : R → R. By elementary calculations one proves Conversely, if f (t) = k j=1 c j exp(ıλ j t), then lim sup t→∞ In the second case it must hold that M{f } = c j * > 0, where c j * corresponds to λ j * = 0.
The mean has the following property, which in general does not hold for functions that have finite sup-or Stepanov norms but are not almost periodic: Lemma 4 Let f : R → R be a real non-negative (a.e.) u.a.p. (/S.a.p.) function. If M{f (t)} = 0, then f (t) = 0 for all (/almost all) t ∈ R.
Proof. The proof of the statement for u.a.p. functions can be found in [3]. We will prove the statement for f being a S.a.p. function. Suppose on the contrary that there exists a non-zero measure set A such that for all t ∈ A we have f (t) > 0. Then there exists a ∈ R such that a+1 a f (u) du > λ for some λ > 0 as follows from Lemma 3. Since f is S.a.p. the set of all the τ such that for all t ∈ R t+1 t |f (u + τ ) − f (u)| < λ 2 is relatively dense. It follows that there exists l λ 2 > 0 such that in any interval I of length l λ 2 we can find τ ∈ I such that a+τ +1 a+τ We can assume that l λ 2 > 1. Then we can find a sequence {τ n } where τ n ∈ [2nl λ 2 , 2(n + 1)l λ ]. Consequently, and we obtain a contradiction.

Remark 1
The assumption that a stimulus function f ∈ L 1 loc (R) allows us also to take as f a linear combination of the Haar wavelets which is more natural for many problems with discontinuous stimulus function.

Displacement map
Definition 6 Let Φ : R → R be a map of the real line. The displacement map Ψ : R → R of Φ is defined as If Φ is a firing map then the displacement map Ψ(t) says how long we have to wait for a next firing if we know that there was a firing at the time t.
Firstly, we will consider the firing map and its displacement for the general case, i.e. for the equation (1). The following observation was first made in [10]: If the function f in (1) is periodic in t (that is, there exists T such that for all x and t we have f (t, x) = f (t + T, x)), then the firing map Φ has periodic displacement. In particular for T = 1 we have Φ(t + 1) = Φ(t) + 1 and thus Φ is a lift of a degree one circle map under the standard projection p : t → exp(2πıt).
Therefore for the models (1) with f periodic in t, the firing map Φ has periodic displacement and induces a degree one circle map (this map is sometimes in the literature referred to as the firing phase map). In this case the tools of the rotation theory ( [12]) can be used to study the properties of Φ. Especially, the rotation number (which is simply the reciprocal of the firing rate, if the unique non-zero firing rate exists) or the rotation intervals are closely related to the so-called phase-locking phenomena ( [4], [10]).
There arises a natural question of whether for the map f almost periodic in t we obtain the firing map with almost periodic displacement. We will tackle this problem for the Perfect Integrator Model.
Firstly, let us consider the case of limit-periodic input: Theorem 3 Let Φ : R → R be the firing map induced by (2). If f : R → R is limit periodic and f (t) > δ for some δ > 0, then the firing map Φ has limit periodic displacement.

From the proof of Theorem 3 follows more general
Corollary 4 Under the assumptions of Theorem 4 the firing map Φ : R → R is uniformly continuous.
Remark 2 Rotation numbers and sets for maps of the real line with almost periodic displacement were investigated by J. Kwapisz in [11]. Observe that Theorems 3 and 4 ensure that the displacement of a firing map is almost periodic if f is so. On the other hand if a given almost periodic function is a firing map then the thesis of main theorems of [11] can be deduced easier by a direct argument. Anyway, it would give an opportunity to compare geometrical definitions of rotation intervals of [12], used in [11], with analytic formulas derived here.

Prospects
As we have already said, our aim was also to establish a framework for study dynamical properties of the firing map Φ and its displacement Ψ.
Now we state a conjecture that the differences of consecutive spikes form an asymptotically semi-periodic, or respectively almost periodic sequence if the input function is so (cf. [2] for definitions of semi-periodic, and almost-periodic sequence).
Let Φ : R → R be the firing map of the equationẋ = f (t) with f ∈ L 1 loc (R) and Ψ : R → R its displacement. For any t ∈ R, we consider a sequence η n (t) = Φ n (t) − Φ n−1 (t) = Ψ(Φ n−1 (t)) Conjecture 1 If f is periodic, correspondingly almost-periodic, then for every t the sequence η n (t) is asymptotically semi-periodic, respectively asymptotically almost-periodic.