Spiky oscillations are characterized by slow-fast dynamics and are observed in excitable media such as neuronal membranes and cardiac cells. In a turbulent reactive flow system, we observe that the heat release rate exhibits self-sustained periodic, spiky oscillations in synchrony with the sinusoidal periodic acoustic pressure oscillations. These self-sustained oscillations are a consequence of thermoacoustic instability, which arises due to a positive feedback between the acoustic and the heat release rate fields. One of the primary mechanisms for fluctuations in the heat release rate is the modulation in the topology of the flame, a thin interface separating the reactants and products. In this work, we explore the dynamics of the space-filling nature of the flame, quantified by its fractal dimension in relation to the spiky heat release rate oscillations in a turbulent reactive flow system. We discover that the periodic oscillatory dynamics in the space-filling nature of the flame lead to periodic, spiky heat release rate oscillations. Based on this result, we show that the spiky oscillations in the heat release rate can be approximated as $e^{sin(\omega t)}$ during the dynamical state of generalized synchronization between the heat release rate and the acoustic pressure oscillations in a turbulent reactive flow system. In synchronization theory, generalized synchronization is characterized by an emergent functional relationship, Φ, between the interacting subsystems. We unravel this emergent functional relation between the heat release rate and the acoustic pressure oscillations in a turbulent reactive flow system. Rather than the specific mathematical expression, the present study unveils that the periodic modulation of the fractal dimension leads to periodic spiky oscillations.